Banach–Tarski paradox

[peteolcott]

We're not saying the sets in the partition are spherical. Their union is a ball, and the union of their images under a bunch of rotational isometries is two balls. The pieces are made using the axiom of Choice, they are not Boreland sets.

Ah so so are trying to get away with fudging the meaning of the words.
If by ball you mean a physically existing object then we are not dealing
with points we are dealing with atoms. If you take away all of the atoms
and divide them into two balls of the same size and shape they now have
half as much mass.
Talking about points on a physically exiting ball is incongruous thus
incoherent. It is either atoms of a ball or points on a sphere. Either
way Banach–Tarski is simply a silly mistake.
Copyright 2018 Pete Olcott

No, I made no reference to physical reality at all, and there was no fudging of the meanings of words. The Banach-Tarski paradox is provable in ZFC. It says that there exists a partition of the solid ball in 3-space into five subsets, four of which have infinitely many points and which are not Borel sets. What you are saying is rubbish.

You would have to point out an actual error of my reasoning for your
assertion that it is rubbish to have any basis what-so-ever.

So you're putting forward a claim to having engaged in reasoning?

Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.

I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.

My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.

On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.

Copyright 2018 Pete Olcott

[peteolcott]

Post by peteolcott

On Saturday, August 18, 2018 at 6:54:18 PM UTC+2,

You would have to point out an actual error of my reasoning
for your assertion that it is rubbish to have any basis
what-so-ever.

So you're putting forward a claim to having engaged in
reasoning?

Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states
that it is possible to decompose a ball into six pieces
which can be reassembled by rigid motions to form two
balls of the same size as the original. The number of
pieces was subsequently reduced to five by Robinson (1947),
although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere
into any subsets and recompose two identical spheres from
these same points.
My *proof* of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) *it is self-evident* that no method for achieving
the desired end-result can possibly exist.

*emphasis added*
"It is self-evident" is not a proof of your claim.

AKA axiomatic. The point is totally proven entirely on the basis
of the meaning of its words.

Your attempt to use the phrase as a proof does prove that
you have not the faintest idea what a proof is, but that's all.

Mutually interlocking semantic specifications. Not the sort of
thing that you would have much knowledge of or experience with.

Post by peteolcott
On the other hand, not having these points georeferenced
would allow the possibility of using the same points twice
to go undetected.

I would suggest that you (PO) go look at the proof in order to
see that points are not being referenced twice, except that
(1) I know you won't do that, and (2) even if you ever did,
you wouldn't understand what you're looking at.

When points are uniquely identified by georeferencing it
is obvious that only a single sphere can possibly exist
with these uniquely identified points.

When-so-ever these points are not uniquely identified then
using a point more than once is indiscernible.

Do you have some other way besides georeferencing that makes
using the same point more than once unequivocally discernible?

I contend that no such system exists, provide a counter-example
proving me wrong on this point.

This whole (Banach–Tarski paradox) only arises because people
are not using my solution (derived in 15 minutes) to totally
abolish the whole issue of: [The Identity of Indiscernibles]
https://plato.stanford.edu/entries/identity-indiscernible/

When the identity of a thing also includes its precise point
in space-time, then two different things are always discernible
and have unique identities.

An imaginary sphere may not have a location in time, yet when
anchored in space has a unique identity. When it is not anchored
in space it lacks a unique identity. Thus the whole idea of
decomposing and recomposing a specific sphere that is not anchored
in space is incorrect because it is impossible to actually
identify any such sphere.

Copyright 2016, 2017, 2018 Pete Olcott

This isn't much different from your argument for the
invalidity of Godel's and Tarski's results on formal
incompleteness and undefinability of truth: You don't
know what is being said, so you make something up.
At that point, it is "self-evident" to you that the thing
you made up is wrong.
I'm not sure I've ever seen you make an argument that is
 "I (PO) don't know what you're saying, so I'll pretend this
 other thing is what you're saying, and show that the
 other thing is wrong."
https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect

[peteolcott]

Post by peteolcott
When the identity of a thing also includes its precise point
in space-time, then two different things are always discernible
and have unique identities.

Your mention of space-time must mean that you're referring
to the actual world. In the actual world, some objects
(fundamental particles, for example) can be different and
also indiscernible.
It's a quantum thing. Suppose we want to know how many
states W the electrons in a piece of silicon crystal _could_
be in, if the total energy of the electrons were U.
 (W is essentially the entropy S of the electrons,
 S = k log W, a very important physical measurement.)
If we count the possible states while assuming that the
electrons are all discernible from each other, _we get_
_the wrong number_ as revealed by our measurements. If we
count while assuming electrons are _not_ discernible,
_we get the right number_ .
For example, suppose we have three orbitals that _could_
accept an electron, and two electrons -- discernible,
we assume, so e1 and e2. How many states are possible?
 [e1][e2][  ],  [e1][  ][e2],  [  ][e1][e2],
 [e2][e1][  ],  [e2][  ][e1],  [  ][e2][e1]
So, six states.
How many states if the electrons are indiscernible?
 [e][e][  ],  [e][  ][e],  [  ][e][e],
 [e][e][  ],  [e][  ][e],  [  ][e][e]
This time, *three states*, because swapping the positions
of indiscernible electrons leaves the state unaltered.
It's an odd thing to be correcting the Creator of the
Universe about, but there you have it. It would be an
understandable mistake to come from one of us non-Creators.
----
Of course, the Banach-Tarski paradox is not about the
actual physical universe, so both your comment and my
rejoinder are irrelevant to that. I just mean to correct
that one mistake in this post, and leave your others
for later.

This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.

When we imagine Banach-Tarski and do not provide some way or another
of uniquely identifying the sphere in question (such as georeferencing)
then we inadvertently conflate one sphere with another and through this
conflation confuse ourselves into thinking that one sphere can
be decomposed into pieces and then subsequently recomposed into two
different spheres.

Yes it is quite unconventional to apply georeferencing to mathematical
objects, new knowledge always tends to be quite unconventional. The
only possible way to eliminate the issue of the identity of (otherwise)
indiscernibles is to specify some set of properties such that a distinction
can always be made between two otherwise indiscernible objects.

It occurred to me less then 15 minutes after first encountering the identity
of indiscernibles through Mitch that otherwise indiscernible objects might
always be made discernible when one considers the property of their point in
time and their points in space.

The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Since these answers delve into religion they are off-topic here because none
of you has a sufficient basis to begin to understand them. Only the first-hand
direct experience of Buddhist enlightenment adequately provides these answers
in a way accessible to the human mind.

Copyright 2016, 2017, 2018 Pete Olcott

[Peter Percival]

Post by peteolcott
The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Schrödinger with his cat intended to refute the Copenhagen
interpretation of qm. It does not illustrate the whole subject.

[peteolcott]

Post by peteolcott
The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Schrödinger with his cat intended to refute the Copenhagen interpretation of qm.  It does not illustrate the whole subject.

By what possible means could a cat actually be simultaneously
alive and dead?

[peteolcott]

Post by peteolcott

Post by peteolcott
The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Schrödinger with his cat intended to refute the Copenhagen interpretation of qm.  It does not illustrate the whole subject.

By what possible means could a cat actually be simultaneously
alive and dead?

What cat?

https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat

So, you might presume we follow the link
and read the entire article.
But, is that Schroedinger's cat?
"Schrödinger's cat is a thought experiment,
sometimes described as a paradox,...".

Is is enormously more than that from the frame-of-reference of Buddhist enlightenment.

So: what cat?
Parastatistics and Bose and Fermi statistics
are various classification of statistics about
particles in probabilistic models of particle
behavior in particle research grounded and
founded with various (or usually, "the Bayesian
for the Bohmian") probability theory.
https://en.wikipedia.org/wiki/David_Bohm
https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory
"In addition to a wavefunction on the space of
all possible configurations, it also postulates
an actual configuration that exists even when
unobserved. The evolution over time of the
configuration (that is, the positions of all
particles or the configuration of all fields)
is defined by the wave function by a guiding
equation. The evolution of the wave function
over time is given by the Schrödinger equation."
So, what cat?
Also, what wave?

[Jeff Barnett]

Post by peteolcott

Post by Peter Percival

Post by peteolcott
The whole subject of quantum mechanics as illustrated by
Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Schrödinger with his cat intended to refute the Copenhagen
interpretation of qm.  It does not illustrate the whole subject.

By what possible means could a cat actually be simultaneously
alive and dead?

By the un-Copenhagen interpretation of QM! Since you don't know logic,
mathematics, linguistics, or programing (I read your silly patent) we
are sure you don't understand physics either: according to the first
sentence, you can't reason, you can't model, you can't communicate, and
you can't express complex processes. All you can do is try to impress -
"Please notice me. Please. Please." We've noticed and the verdict is in:
dunce!

Now please let these newsgroups go back to being semi-useful and you
take a long rest. Please. Please.

[peteolcott]

Post by Jeff Barnett

Post by peteolcott

Post by peteolcott
The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Schrödinger with his cat intended to refute the Copenhagen interpretation of qm.  It does not illustrate the whole subject.

By what possible means could a cat actually be simultaneously
alive and dead?

By the un-Copenhagen interpretation of QM!

According to the Copenhagen interpretation, physical
systems generally do not have definite properties prior
to being measured, and quantum mechanics can only
predict the probabilities that measurements will produce
certain results. The act of measurement affects the system,
causing the set of probabilities to reduce to only
one of the possible values immediately after the
measurement. This feature is known as wave function collapse.

Most people do not realize that this "answer" only dodges the question.

What assumptions regarding the fundamental nature of reality
would support the above behavior of physical systems?

Ah that never occurred to you, as I would have guessed.
All questions that do not have answers that can be looked
up do not count as worthy questions?

Post by Jeff Barnett
Since you don't know logic, mathematics, linguistics, or programing (I read your silly patent) we are sure you don't understand physics either: according to the first sentence, you can't reason, you can't model,
you can't communicate, and you can't express complex processes. All you can do is try to impress - "Please notice me. Please. Please." We've noticed and the verdict is in: dunce!
Now please let these newsgroups go back to being semi-useful and you take a long rest. Please. Please.

[peteolcott]

Post by peteolcott

Post by Jeff Barnett

Post by peteolcott
By what possible means could a cat actually be
simultaneously alive and dead?

By the un-Copenhagen interpretation of QM!

According to the Copenhagen interpretation, physical
systems generally do not have definite properties prior
to being measured, and quantum mechanics can only
predict the probabilities that measurements will produce
certain results. The act of measurement affects the system,
causing the set of probabilities to reduce to only
one of the possible values immediately after the
measurement. This feature is known as wave function collapse.
Most people do not realize that this "answer" only dodges
the question.
What assumptions regarding the fundamental nature of reality
would support the above behavior of physical systems?

There are assumptions about the _classical_ view of reality
that the _quantum_ view of reality contradicts. It turns
out that, with regard to reality, the classical view is wrong.
 (The classical view is _approximately_ correct for a wide

This is totally incorrect.
We are fooled into thinking that:
The classical view is _approximately_ correct.

It is not actually correct at all, even though reality
really seems to act as if the classical view is correct,
this is only how things seem to be, and not at all how
they actually are.

Like I already said short of full Buddhist enlightenment
it is totally inaccessible to the human mind.

 range of circumstances in which we, inhabiting our smaller-
 -than-galaxies, larger-than-atoms bodies, find ourselves.
 This is why we find _reality_ outside our usual experience
 (eg, quantum mechanics experiments) to be counter-intuitive.)
Consider Bell's theorem.
https://en.wikipedia.org/wiki/Bell%27s_theorem#Importance
<wiki>
 The title of Bell's seminal article refers to the 1935
 paper by Einstein, Podolsky and Rosen that challenged the
 completeness of quantum mechanics. In his paper, Bell
 started from the same two assumptions as did EPR, namely
 (i) reality (that microscopic objects have real properties
 determining the outcomes of quantum mechanical measurements),
 and (ii) locality (that reality in one location is not
 influenced by measurements performed simultaneously at a
 distant location). Bell was able to derive from those two
 assumptions an important result, namely Bell's inequality.
 The theoretical (and later experimental) violation of this
 inequality implies that at least one of the two assumptions
 must be false.
</wiki>

Post by peteolcott
Ah that never occurred to you, as I would have guessed.
All questions that do not have answers that can be looked
up do not count as worthy questions?

I think you claimed elsewhere that you spent 15 minutes
thinking about this. I doubt you spent that much time, but
even so: 15 minutes? Did you even do one Google search
on your topic?
You _guessed_ that no one has looked at these questions,
and that _guess_ became your "Truth". The same for Godel.
The same for Tarski. And for Cantor. And for every other
question you "work" on. I doubt you see anything wrong
with that.
https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect

[Jim Burns]

Post by peteolcott

Post by peteolcott

Post by Jeff Barnett

Post by peteolcott
By what possible means could a cat actually be
simultaneously alive and dead?

By the un-Copenhagen interpretation of QM!

According to the Copenhagen interpretation, physical
systems generally do not have definite properties prior
to being measured, and quantum mechanics can only
predict the probabilities that measurements will produce
certain results. The act of measurement affects the
system, causing the set of probabilities to reduce to
only one of the possible values immediately after the
measurement. This feature is known as wave function
collapse.
Most people do not realize that this "answer" only dodges
the question.
What assumptions regarding the fundamental nature of reality
would support the above behavior of physical systems?

There are assumptions about the _classical_ view of
reality that the _quantum_ view of reality contradicts.
It turns out that, with regard to reality, the classical
view is wrong.
  (The classical view is _approximately_ correct for a wide

This is totally incorrect.
The classical view is _approximately_ correct.
It is not actually correct at all,

You:
<PO>

Post by peteolcott
As I say too many times to count, I ONLY refer

</PO>

So, that's bullshit.

<snip blahblahblah>

[Peter T. Daniels]

Post by peteolcott

Post by peteolcott
The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Schrödinger with his cat intended to refute the Copenhagen interpretation of qm.  It does not illustrate the whole subject.

By what possible means could a cat actually be simultaneously
alive and dead?

Watch *The Big Bang Theory*. Penny will explain it to you.

[peteolcott]

Post by Peter T. Daniels

Post by peteolcott

Post by peteolcott
The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Schrödinger with his cat intended to refute the Copenhagen interpretation of qm.  It does not illustrate the whole subject.

By what possible means could a cat actually be simultaneously
alive and dead?

Watch *The Big Bang Theory*. Penny will explain it to you.

As I expected everyone failed to provide the correct answer.

[Peter Percival]

Post by peteolcott

Post by Peter Percival

Post by peteolcott
The whole subject of quantum mechanics as illustrated by
Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Schrödinger with his cat intended to refute the Copenhagen
interpretation of qm.  It does not illustrate the whole subject.

By what possible means could a cat actually be simultaneously
alive and dead?

Schrödinger with his cat intended to refute the Copenhagen
interpretation of qm.

[peteolcott]

Post by peteolcott

Post by peteolcott
The whole subject of quantum mechanics as illustrated by Schrödinger's cat
has its ultimate ground of being in the actual true nature of reality as
opposed to common misconceptions of this nature of reality.

Schrödinger with his cat intended to refute the Copenhagen interpretation of qm.  It does not illustrate the whole subject.

By what possible means could a cat actually be simultaneously
alive and dead?

Schrödinger with his cat intended to refute the Copenhagen interpretation of qm.

"physical systems generally do not have
definite properties prior to being measured"

What definition of reality makes that work?

[Peter T. Daniels]

Post by peteolcott
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.

Nothing to do with human language. There's no such thing as "formal semantics
of linguistics."

[DKleinecke]

Post by Peter T. Daniels

Post by peteolcott
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.

Nothing to do with human language. There's no such thing as "formal semantics
of linguistics."

There does seem to be a group of people who try to add
semantics to Chomskian formalism and call the result
"formal semantics". Montague started it and it doesn't
seem to have died out yet.

[peteolcott]

Post by DKleinecke

Post by Peter T. Daniels

Post by peteolcott
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.

Nothing to do with human language. There's no such thing as "formal semantics
of linguistics."

There does seem to be a group of people who try to add
semantics to Chomskian formalism and call the result
"formal semantics". Montague started it and it doesn't
seem to have died out yet.

http://www.cyc.com/documentation/ontologists-handbook/writing-efficient-cycl/cycl-representation-choices/
Since you already acknowledged the validity of the above why are you contradicting yourself now?

[DKleinecke]

Post by peteolcott

Post by DKleinecke

Post by Peter T. Daniels

Post by peteolcott
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.

Nothing to do with human language. There's no such thing as "formal semantics
of linguistics."

There does seem to be a group of people who try to add
semantics to Chomskian formalism and call the result
"formal semantics". Montague started it and it doesn't
seem to have died out yet.

http://www.cyc.com/documentation/ontologists-handbook/writing-efficient-cycl/cycl-representation-choices/
Since you already acknowledged the validity of the above why are you contradicting yourself now?

I don't see any "formal semantics" here.

I agree more-or-less with Davidson and can see how Cycorp
is trying to implement Davidson's insight. But what does
that have to do with your theories?

[peteolcott]

Post by DKleinecke

Post by peteolcott

Post by DKleinecke

Post by Peter T. Daniels

Post by peteolcott
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.

Nothing to do with human language. There's no such thing as "formal semantics
of linguistics."

There does seem to be a group of people who try to add
semantics to Chomskian formalism and call the result
"formal semantics". Montague started it and it doesn't
seem to have died out yet.

http://www.cyc.com/documentation/ontologists-handbook/writing-efficient-cycl/cycl-representation-choices/
Since you already acknowledged the validity of the above why are you contradicting yourself now?

I don't see any "formal semantics" here.
I agree more-or-less with Davidson and can see how Cycorp
is trying to implement Davidson's insight. But what does
that have to do with your theories?

explained in the other reply.

[peteolcott]

Post by Peter T. Daniels

Post by peteolcott
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.

Nothing to do with human language. There's no such thing as "formal semantics
of linguistics."

https://academic.oup.com/jos
About the journal
Journal of Semantics covers all areas in the study
of meaning, with a focus on formal and experimental
methods. It welcomes submissions on semantics,
pragmatics, the syntax/semantics interface,
cross-linguistic semantics, experimental studies of
meaning, and semantically informed philosophy of language.

When anyone repeats the same false claim after
being repeatedly corrected it must be dishonestly.

[Peter T. Daniels]

Post by peteolcott

Post by Peter T. Daniels

Post by peteolcott
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.

Nothing to do with human language. There's no such thing as "formal semantics
of linguistics."

https://academic.oup.com/jos
About the journal
Journal of Semantics covers all areas in the study
of meaning, with a focus on formal and experimental
methods. It welcomes submissions on semantics,
pragmatics, the syntax/semantics interface,
cross-linguistic semantics, experimental studies of
meaning, and semantically informed philosophy of language.
When anyone repeats the same false claim after
being repeatedly corrected it must be dishonestly.

I don't know what else that journal publishes, but if it publishes anything
like what you display here, then it is not publishing about the semantics
of human language in those articles.

[peteolcott]

Post by Peter T. Daniels

Post by peteolcott

Post by Peter T. Daniels

Post by peteolcott
This is related to sci.lang in that it further elaborates the mathematics
of semantics which is the broader subject of the mathematics of the
meaning of words, AKA formal semantics of linguistics.

Nothing to do with human language. There's no such thing as "formal semantics
of linguistics."

https://academic.oup.com/jos
About the journal
Journal of Semantics covers all areas in the study
of meaning, with a focus on formal and experimental
methods. It welcomes submissions on semantics,
pragmatics, the syntax/semantics interface,
cross-linguistic semantics, experimental studies of
meaning, and semantically informed philosophy of language.
When anyone repeats the same false claim after
being repeatedly corrected it must be dishonestly.

I don't know what else that journal publishes, but if it publishes anything
like what you display here, then it is not publishing about the semantics
of human language in those articles.

Yes of course everyone knows that [linguistics] only refer
to algebra and never refers to natural human language. That
is why the journal uses the term [linguistics] in its description.

[wugi]

Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.

Even if formulated in a more challenging way than the following
examples, I see hardly a difference with the fact that any line segment
is equipotent (or what's it called) to any other line segment, yes to
any surface, to any volume, and so on: R ~ R^n.

So let's take two concentric circles with radii r and 2r. Circle r can
self-evidently be transformed onto circle 2r. Each point of circle r
finds its own place on circle 2r by a simple radial transfer (and vice
versa). Afterwards, from the circle 2r curve you can make two new
circles with radius r.

QED

[peteolcott]

Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.

Even if formulated in a more challenging way than the following examples, I see hardly a difference with the fact that any line segment is equipotent (or what's it called) to any other line segment, yes to any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r can self-evidently be transformed onto circle 2r. Each point of circle r finds its own place on circle 2r by a simple radial transfer (and vice versa). Afterwards, from the circle 2r curve
you can make two new circles with radius r.
QED

This whole (Banach–Tarski paradox) only arises because people
are not using my solution (derived in 15 minutes) to totally
abolish the whole issue of: [The Identity of Indiscernibles]
https://plato.stanford.edu/entries/identity-indiscernible/

When the identity of a thing also includes its precise point
in space-time, then two different things are always discernible
and have unique identities.

An imaginary sphere may not have a location in time, yet when
anchored in space has a unique identity. When it is not anchored
in space it lacks a unique identity. Thus the whole idea of
decomposing and recomposing a specific sphere that is not anchored
in space is incorrect because it is impossible to actually
identify any such sphere.

Copyright 2016, 2017, 2018 Pete Olcott

[Peter Percival]

Post by peteolcott

Post by wugi

Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.

Even if formulated in a more challenging way than the following
examples, I see hardly a difference with the fact that any line
segment is equipotent (or what's it called) to any other line segment,
yes to any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r can
self-evidently be transformed onto circle 2r. Each point of circle r
finds its own place on circle 2r by a simple radial transfer (and vice
versa). Afterwards, from the circle 2r curve you can make two new
circles with radius r.
QED

This whole (Banach–Tarski paradox) only arises because people
are not using my solution (derived in 15 minutes) to totally
abolish the whole issue of: [The Identity of Indiscernibles]
https://plato.stanford.edu/entries/identity-indiscernible/
When the identity of a thing also includes its precise point
in space-time

What have points in space-time got to do with the Banach–Tarski paradox?
It seems that you do not know what mathematicians mean by "sphere".

Post by peteolcott
, then two different things are always discernible
and have unique identities.
An imaginary sphere may not have a location in time, yet when
anchored in space has a unique identity. When it is not anchored
in space it lacks a unique identity. Thus the whole idea of
decomposing and recomposing a specific sphere that is not anchored
in space is incorrect because it is impossible to actually
identify any such sphere.
Copyright 2016, 2017, 2018 Pete Olcott

[peteolcott]

Post by peteolcott

Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.

Even if formulated in a more challenging way than the following examples, I see hardly a difference with the fact that any line segment is equipotent (or what's it called) to any other line segment, yes to any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r can self-evidently be transformed onto circle 2r. Each point of circle r finds its own place on circle 2r by a simple radial transfer (and vice versa). Afterwards, from the circle 2r
curve you can make two new circles with radius r.
QED

This whole (Banach–Tarski paradox) only arises because people
are not using my solution (derived in 15 minutes) to totally
abolish the whole issue of: [The Identity of Indiscernibles]
https://plato.stanford.edu/entries/identity-indiscernible/
When the identity of a thing also includes its precise point
in space-time

What have points in space-time got to do with the Banach–Tarski paradox?  It seems that you do not know what mathematicians mean by "sphere".

I augmented the conception of a sphere such that an individual
sphere can be uniquely identified and thus not inadvertently
conflated with other different spheres having the exact same size.

When I add this required extra degree of discernment, Banach–Tarski
cannot slip through the cracks of vagueness, thus ceases to exist.

Copyright 2018 Pete Olcott

Post by peteolcott
, then two different things are always discernible
and have unique identities.
An imaginary sphere may not have a location in time, yet when
anchored in space has a unique identity. When it is not anchored
in space it lacks a unique identity. Thus the whole idea of
decomposing and recomposing a specific sphere that is not anchored
in space is incorrect because it is impossible to actually
identify any such sphere.
Copyright 2016, 2017, 2018 Pete Olcott

[Peter Percival]

Post by peteolcott

Post by Peter Percival

Post by peteolcott

Post by wugi

Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.

Even if formulated in a more challenging way than the following
examples, I see hardly a difference with the fact that any line
segment is equipotent (or what's it called) to any other line
segment, yes to any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r
can self-evidently be transformed onto circle 2r. Each point of
circle r finds its own place on circle 2r by a simple radial
transfer (and vice versa). Afterwards, from the circle 2r curve you
can make two new circles with radius r.
QED

This whole (Banach–Tarski paradox) only arises because people
are not using my solution (derived in 15 minutes) to totally
abolish the whole issue of: [The Identity of Indiscernibles]
https://plato.stanford.edu/entries/identity-indiscernible/
When the identity of a thing also includes its precise point
in space-time

What have points in space-time got to do with the Banach–Tarski
paradox?  It seems that you do not know what mathematicians mean by
"sphere".

I augmented the conception of a sphere

That's very obliging of you. The Banach–Tarski paradox is about
mathematical spheres, it is not about your augmentation.

Post by peteolcott
such that an individual
sphere can be uniquely identified and thus not inadvertently
conflated with other different spheres having the exact same size.
When I add this required extra degree of discernment, Banach–Tarski
cannot slip through the cracks of vagueness, thus ceases to exist.
Copyright 2018 Pete Olcott

Post by Peter Percival

Post by peteolcott
, then two different things are always discernible
and have unique identities.
An imaginary sphere may not have a location in time, yet when
anchored in space has a unique identity. When it is not anchored
in space it lacks a unique identity. Thus the whole idea of
decomposing and recomposing a specific sphere that is not anchored
in space is incorrect because it is impossible to actually
identify any such sphere.
Copyright 2016, 2017, 2018 Pete Olcott

[Mike Terry]

Post by wugi

Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.

Even if formulated in a more challenging way than the following
examples, I see hardly a difference with the fact that any line segment
is equipotent (or what's it called) to any other line segment, yes to
any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r can
self-evidently be transformed onto circle 2r. Each point of circle r
finds its own place on circle 2r by a simple radial transfer (and vice
versa). Afterwards, from the circle 2r curve you can make two new
circles with radius r.
QED

True, but here you're partitioning the cirlce into infinitely many
partitions, which is missing a key point.

Try again, but dividing the circle into only a finite number of pieces!

Regards,
Mike.

[peteolcott]

Post by wugi

Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.

Even if formulated in a more challenging way than the following
examples, I see hardly a difference with the fact that any line segment
is equipotent (or what's it called) to any other line segment, yes to
any surface, to any volume, and so on: R ~ R^n.
So let's take two concentric circles with radii r and 2r. Circle r can
self-evidently be transformed onto circle 2r. Each point of circle r
finds its own place on circle 2r by a simple radial transfer (and vice
versa). Afterwards, from the circle 2r curve you can make two new
circles with radius r.
QED

True, but here you're partitioning the cirlce into infinitely many partitions, which is missing a key point.
Try again, but dividing the circle into only a finite number of pieces!
Regards,
Mike.

Which cannot possibly be recomposed into two identical spheres if every point of these original two pieces has been georeferenced.

If every point in the original sphere had not been georeferenced then the original sphere would have never been uniquely identified thus even talking about it would be incorrect.

[Franz Gnaedinger]

Copyright 2018 Pete Olcott

Peter Olcott knows the absolute and complete and total truth, he is the author
of life and creator of life, he has hundred reasons to assume that he is God,
he creates our future minds in order that we can go on existing, and he is the
one Creator of the Universe (claims he made in sci.lang and which entitle him
to start ever more threads). The self-declared author of life imposes a lifeless
language on us. He castrates language in the name of mathematical logic, and
mathematical logic by dismissing proven theorems. Goedel was wrong, Turing was
wrong, Allgod is right. He calls the understanding of Goedel's proven theorems
(a pleonasm I can't repeat often enough) a religious conviction. Meaning heresy.
He is Allgod and holds a copyright on the Truth. We must believe in Him.

[Peter Percival]

Post by peteolcott

We're not saying the sets in the partition are spherical. Their
union is a ball, and the union of their images under a bunch of
rotational isometries is two balls. The pieces are made using the
axiom of Choice, they are not Boreland sets.

Ah so so are trying to get away with fudging the meaning of the words.
If by ball you mean a physically existing object then we are not dealing
with points we are dealing with atoms. If you take away all of the atoms
and divide them into two balls of the same size and shape they now have
half as much mass.
Talking about points on a physically exiting ball is incongruous thus
incoherent. It is either atoms of a ball or points on a sphere. Either
way Banach–Tarski is simply a silly mistake.
Copyright 2018 Pete Olcott

No, I made no reference to physical reality at all, and there was no
fudging of the meanings of words. The Banach-Tarski paradox is
provable in ZFC. It says that there exists a partition of the solid
ball in 3-space into five subsets, four of which have infinitely
many points and which are not Borel sets. What you are saying is
rubbish.

You would have to point out an actual error of my reasoning for your
assertion that it is rubbish to have any basis what-so-ever.

So you're putting forward a claim to having engaged in reasoning?

Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced

What do you mean by "georeferenced"? It is not a word I have come
across before.

Post by peteolcott
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Copyright 2018 Pete Olcott

[peteolcott]

Post by peteolcott

We're not saying the sets in the partition are spherical. Their union is a ball, and the union of their images under a bunch of rotational isometries is two balls. The pieces are made using the axiom of Choice, they are not Boreland sets.

Ah so so are trying to get away with fudging the meaning of the words.
If by ball you mean a physically existing object then we are not dealing
with points we are dealing with atoms. If you take away all of the atoms
and divide them into two balls of the same size and shape they now have
half as much mass.
Talking about points on a physically exiting ball is incongruous thus
incoherent. It is either atoms of a ball or points on a sphere. Either
way Banach–Tarski is simply a silly mistake.
Copyright 2018 Pete Olcott

No, I made no reference to physical reality at all, and there was no fudging of the meanings of words. The Banach-Tarski paradox is provable in ZFC. It says that there exists a partition of the solid ball in 3-space into five subsets, four of which
have infinitely many points and which are not Borel sets. What you are saying is rubbish.

You would have to point out an actual error of my reasoning for your
assertion that it is rubbish to have any basis what-so-ever.

So you're putting forward a claim to having engaged in reasoning?

Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced

What do you mean by "georeferenced"?  It is not a word I have come across before.

I already explained that:

The center of a one inch diameter sphere is exactly ten miles above
the center of the north pole.

Without this degree of specificity the sphere is never uniquely
identified thus indiscernible from other spheres.

When we decompose this sphere into any parts and recompose them it is
impossible to create two spheres, thus Banach–Tarski has been fully refuted.

Banach–Tarski only exists because no one ever bothered to uniquely
identify the specific sphere in question, thus got confused and made a
copy of the sphere without realizing that it was only a copy and not
the original sphere at all.

The identity of otherwise indiscernibles is always uniquely established
if anchored in points in space and/or a point in time as appropriate.
In Banach–Tarski we anchor the sphere in points in space.

Copyright 2016, 2017, 2018 Pete Olcott

Post by peteolcott
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Copyright 2018 Pete Olcott

[Peter Percival]

Post by peteolcott

Post by peteolcott

We're not saying the sets in the partition are spherical. Their
union is a ball, and the union of their images under a bunch of
rotational isometries is two balls. The pieces are made using
the axiom of Choice, they are not Boreland sets.

Ah so so are trying to get away with fudging the meaning of the words.
If by ball you mean a physically existing object then we are not dealing
with points we are dealing with atoms. If you take away all of the atoms
and divide them into two balls of the same size and shape they now have
half as much mass.
Talking about points on a physically exiting ball is incongruous thus
incoherent. It is either atoms of a ball or points on a sphere. Either
way Banach–Tarski is simply a silly mistake.
Copyright 2018 Pete Olcott

No, I made no reference to physical reality at all, and there was
no fudging of the meanings of words. The Banach-Tarski paradox is
provable in ZFC. It says that there exists a partition of the
solid ball in 3-space into five subsets, four of which have
infinitely many points and which are not Borel sets. What you are
saying is rubbish.

You would have to point out an actual error of my reasoning for your
assertion that it is rubbish to have any basis what-so-ever.

So you're putting forward a claim to having engaged in reasoning?

Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is that when these points are georeferenced

What do you mean by "georeferenced"?  It is not a word I have come across before.

The center of a one inch diameter sphere is exactly ten miles above
the center of the north pole.

These physical things have nothing to do with the Banach–Tarski paradox.
Do you even know what a sphere is to a mathematician? It's not a
rhetorical question, I'd really like to know if you know.

Post by peteolcott
Without this degree of specificity the sphere is never uniquely
identified thus indiscernible from other spheres.
When we decompose this sphere into any parts and recompose them it is
impossible to create two spheres, thus Banach–Tarski has been fully refuted.

Read Banach and Tarski's paper. Where is the first error?

Post by peteolcott
Banach–Tarski only exists because no one ever bothered to uniquely
identify the specific sphere in question, thus got confused and made a

Because there is no "the" sphere. The theorem is true of all
mathematical spheres.

Post by peteolcott
copy of the sphere without realizing that it was only a copy and not
the original sphere at all.
The identity of otherwise indiscernibles is always uniquely established
if anchored in points in space and/or a point in time as appropriate.
In Banach–Tarski we anchor the sphere in points in space.

Time and space are irrelevant.

You're a bit of an idiot, aren't you? And you have no sense of shame.

[peteolcott]

Post by peteolcott

What do you mean by "georeferenced"?  It is not a word
I have come across before.

The center of a one inch diameter sphere is exactly ten
miles above the center of the north pole.
Without this degree of specificity the sphere is never
uniquely identified thus indiscernible from other spheres.

Let the center of a one-inch-diameter sphere S be ten miles
above the north pole. _Rigidly translate_ the points in
sphere S to another location such that the new center of S
is ten miles above the _south_ pole.

It thus ceases to be the original sphere at all.
If by what-so-ever means you end up with more than
one sphere, you simply duplicated the original sphere.

As soon as the sphere is moved from its original location
it loses its identity and is no longer the original sphere
at all.

Think of it as the same idea as the Cantor's cardinality proof.
We have an original sphere that has its own set of unique
points. A set of points that was previously not mapped to
any other points after decomposition and recomposition becomes
mapped to a whole new set of points that did not previously
exist. This proves that we really only just copied the sphere
without realizing it.

Copyright 2018 Pete Olcott

Is that impossible to do? Is that what you're claiming?
Because that is the sort of thing that the Banach-Tarski
theorem asks us to do, only instead of rigidly translating
a whole sphere, pieces are rigidly translated, to two
different locations, and there assembled -- rigidly --
into two spheres each the same size as the original sphere.
Go ahead and give the centers of the spheres whatever
latitude, longitude, and altitude you like. That's utterly
irrelevant.

Post by peteolcott
When we decompose this sphere into any parts and recompose
them it is impossible to create two spheres,

You don't say _why_ it is impossible.
You don't seem to be aware that you are not saying _why_
it is impossible.

Post by peteolcott
thus Banach–Tarski has been fully  refuted.
Banach–Tarski only exists because no one ever bothered to
uniquely identify the specific sphere in question, thus
got confused and made a copy of the sphere without realizing
that it was only a copy and not the original sphere at all.
The identity of otherwise indiscernibles is always uniquely
established if anchored in points in space and/or a point in
time as appropriate. In Banach–Tarski we anchor the sphere
in points in space.

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
<wiki>
 The reason the Banach-Tarski theorem is called a paradox is
 that it contradicts basic geometric intuition. "Doubling the
 ball" by dividing it into parts and moving them around by
 rotations and translations, without any stretching, bending,
 or adding new points, seems to be impossible, since all these
 operations ought, intuitively speaking, to preserve the volume.
 The intuition that such operations preserve volumes is not
 mathematically absurd and it is even included in the formal
 definition of volumes. However, this is not applicable here
 because in this case it is impossible to define the volumes
 of the considered subsets. Reassembling them reproduces a
 volume, which happens to be different from the volume at
 the start.
</wiki>

[Jim Burns]

Post by peteolcott

Post by peteolcott

What do you mean by "georeferenced"?  It is not a word
I have come across before.

The center of a one inch diameter sphere is exactly ten
miles above the center of the north pole.
Without this degree of specificity the sphere is never
uniquely identified thus indiscernible from other spheres.

Let the center of a one-inch-diameter sphere S be ten miles
above the north pole. _Rigidly translate_ the points in
sphere S to another location such that the new center of S
is ten miles above the _south_ pole.

It thus ceases to be the original sphere at all.

Apparently you got the idea from somewhere that
Banach-Tarski claim is that the two resulting spheres are
in the same location as the one original sphere.

Have you considered finding out what the Banach-Tarski
theorem is? (BIG spoiler: It's not what you think it is.)

[peteolcott]

Post by Jim Burns

Post by peteolcott

Post by peteolcott

What do you mean by "georeferenced"?  It is not a word
I have come across before.

The center of a one inch diameter sphere is exactly ten
miles above the center of the north pole.
Without this degree of specificity the sphere is never
uniquely identified thus indiscernible from other spheres.

Let the center of a one-inch-diameter sphere S be ten miles
above the north pole. _Rigidly translate_ the points in
sphere S to another location such that the new center of S
is ten miles above the _south_ pole.

It thus ceases to be the original sphere at all.

Apparently you got the idea from somewhere that
Banach-Tarski claim is that the two resulting spheres are
in the same location as the one original sphere.
Have you considered finding out what the Banach-Tarski
theorem is? (BIG spoiler: It's not what you think it is.)

Think of it as the same idea as the Cantor's cardinality proof.
We have an original sphere that has its own set of unique
points. A set of points that was previously not mapped to
any other points after decomposition and recomposition becomes
mapped to a whole new set of points that did not previously
exist. This proves that we really only just copied the sphere
without realizing it.

Copyright 2018 Pete Olcott

[peteolcott]

Post by peteolcott

Post by Jim Burns

Post by peteolcott

Post by peteolcott

What do you mean by "georeferenced"?  It is not a word
I have come across before.

The center of a one inch diameter sphere is exactly ten
miles above the center of the north pole.
Without this degree of specificity the sphere is never
uniquely identified thus indiscernible from other spheres.

Let the center of a one-inch-diameter sphere S be ten miles
above the north pole. _Rigidly translate_ the points in
sphere S to another location such that the new center of S
is ten miles above the _south_ pole.

It thus ceases to be the original sphere at all.

Apparently you got the idea from somewhere that
Banach-Tarski claim is that the two resulting spheres are
in the same location as the one original sphere.
*Have you considered finding out what the Banach-Tarski*
*theorem is* ? (BIG spoiler: It's not what you think it is.)

Think of it as the same idea as the Cantor's cardinality
proof. We have an original sphere that has its own set of
unique points. A set of points that was previously not
mapped to any other points after decomposition and
recomposition becomes mapped to a whole new set of points
that did not previously exist. This proves that we really
only just copied the sphere without realizing it.

No. Just no.
I point out one misconception, and you respond with six
more misconceptions. I point out one of the new
misconceptions, and you return another six misconceptions.
With each of my answers, you diverge further and further
from reality. So, just no.
*Have you considered finding out what the Cantor's*
*cardinality proof is* ?

It is a mathematical mapping between set of points
on a number line.

I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

It is not merely decomposing a single sphere and
recomposing this same sphere into two identical
spheres.

It is copying the points from one sphere to another
in such a way that it is not apparent that it is
copying anything.

Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.

Copyright 2018 Pete Olcott

Copyright 2018 Pete Olcott

[peteolcott]

Post by peteolcott
I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

No you haven't. Try watching this:

Maybe even you can understand it if you don't have to read.
EFQ

if Banach-Tarski asserts it decomposes a sphere into
parts and then recompose two spheres from these same
parts then I have proven that Banach-Tarski is asserting
a falsehood.

The first two minutes of the video confirm that my
understanding of the Banach-Tarski assertion is correct.

The details after this assertion are moot as long as
I prove that the assertion itself is impossible.

I take exactly this same approach on the 1931 GIT. I do
not need to understand any detail of the 1931 GIT to
utterly refute it as long as I prove that its conclusion
is impossible.

Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.

Copyright 2016, 2017, 2018 Pete Olcott

[peteolcott]

Post by peteolcott

Post by peteolcott
I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

No you haven't.  Try watching this: http://youtu.be/s86-Z-CbaHA
Maybe even you can understand it if you don't have to read.
EFQ

if Banach-Tarski asserts it decomposes a sphere into
parts and then recompose two spheres

each identical to the original

Post by peteolcott
from these same
parts then I have proven that Banach-Tarski is asserting
a falsehood.
The first two minutes of the video confirm that my
understanding of the Banach-Tarski assertion is correct.
The details after this assertion are moot as long as
I prove that the assertion itself is impossible.
I take exactly this same approach on the 1931 GIT. I do
not need to understand any detail of the 1931 GIT to
utterly refute it as long as I prove that its conclusion
is impossible.
Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.
Copyright 2016, 2017, 2018 Pete Olcott

[peteolcott]

Post by peteolcott

Post by peteolcott

Post by peteolcott
I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

No you haven't.  Try watching this: http://youtu.be/s86-Z-CbaHA
Maybe even you can understand it if you don't have to read.

if Banach-Tarski asserts it decomposes a sphere into
parts and then recompose two spheres

each identical to the original

They are "identical" in that they have the same number of points, the same volume, the same surface area, etc. "Equivalent" is a less loaded term than "identical".
EFQ

What you are not getting is the inherent impossibility
of creating a bijective mapping from a set of points
that did not previously exist to the original set of
points by merely rearranging the original set of points.

You and up with a bijective mapping between two spheres
whereas only the set of points of one of these spheres
originally existed.

Copyright Pete Olcott 2018

[peteolcott]

Post by peteolcott

Post by peteolcott

Post by peteolcott
if Banach-Tarski asserts it decomposes a sphere into
parts and then recompose two spheres

each identical to the original

They are "identical" in that they have the same number of points, the same volume, the same surface area, etc. "Equivalent" is a less loaded term than "identical".

What you are not getting is the inherent impossibility

It's not impossible, so why would I want to "get" that it's impossible? That would be stupid.
I bet you think that the set of integers is twice as large as the set of even integers.
EFQ

Here it is at a higher level of abstraction:
Olcott's KEY hypothesis:
All paradox necessarily has hidden error at its core.
You may not have the capacity to understand this.

Copyright 2016, 2017, 2018 Pete Olcott

[peteolcott]

Post by peteolcott

Post by peteolcott

Post by peteolcott
if Banach-Tarski asserts it decomposes a sphere into
parts and then recompose two spheres

each identical to the original

They are "identical" in that they have the same number of points, the same volume, the same surface area, etc. "Equivalent" is a less loaded term than "identical".

What you are not getting is the inherent impossibility

It's not impossible, so why would I want to "get" that it's impossible? That would be stupid.
I bet you think that the set of integers is twice as large as the set of even integers.
EFQ

Euh, the asymptotic density of the even integers
(in the integers) is one half.
https://en.wikipedia.org/wiki/Schnirelmann_density
This asymptotic or Schnirelmann density is a usual
fact about the numbers and in terms of "measures"
as it were, of the "size" of things.
We can point to F. Katz and OUTPACING also as about
how proper subsets of a set are "smaller".
https://arxiv.org/abs/math/0106100
Of course everyone here knows about trans-finite
cardinals as about "infinite counting", and about
how Galileo showed the integers biject, for example,
with the squares.

OK then the Banach–Tarski paradox doubles the number of points
by simply rearranging them without copying them. Since this would
be impossible therefore Banach–Tarski is impossible.

[Peter Percival]

Post by peteolcott

On Tuesday, August 21, 2018 at 11:01:05 PM UTC-7,

Post by peteolcott

Post by peteolcott

Post by peteolcott
if Banach-Tarski asserts it decomposes a sphere into
parts and then recompose two spheres

each identical to the original

They are "identical" in that they have the same number of points,
the same volume, the same surface area, etc. "Equivalent" is a less
loaded term than "identical".

What you are not getting is the inherent impossibility

It's not impossible, so why would I want to "get" that it's
impossible? That would be stupid.
I bet you think that the set of integers is twice as large as the set of even integers.
EFQ

Euh, the asymptotic density of the even integers
(in the integers) is one half.
https://en.wikipedia.org/wiki/Schnirelmann_density
This asymptotic or Schnirelmann density is a usual
fact about the numbers and in terms of "measures"
as it were, of the "size" of things.
We can point to F. Katz and OUTPACING also as about
how proper subsets of a set are "smaller".
https://arxiv.org/abs/math/0106100
Of course everyone here knows about trans-finite
cardinals as about "infinite counting", and about
how Galileo showed the integers biject, for example,
with the squares.

OK then the Banach–Tarski paradox doubles the number of points
by simply rearranging them without copying them. Since this would
be impossible therefore Banach–Tarski is impossible.

Find a proof of the Banach–Tarski paradox and read it. You won't
because you won't understand it. Can you learn something from your not
understanding it?

You are one of those people who is so stupid that you don't know how
stupid you are. I'm not saying you're stupid because you don't
understand the Banach–Tarski paradox, most intelligent people have never
heard of it. The reason I say you're stupid is because you think it is
reasonable to comment on the Banach–Tarski paradox (and Gödel's
incompleteness theorem and ...) from a position of ignorance, what is
more a position of ignorance that you are determined to maintain.

You people have no sense of shame.

[peteolcott]

Post by peteolcott

Post by peteolcott

Post by peteolcott
if Banach-Tarski asserts it decomposes a sphere into
parts and then recompose two spheres

each identical to the original

They are "identical" in that they have the same number of points, the same volume, the same surface area, etc. "Equivalent" is a less loaded term than "identical".

What you are not getting is the inherent impossibility

It's not impossible, so why would I want to "get" that it's impossible? That would be stupid.
I bet you think that the set of integers is twice as large as the set of even integers.

Euh, the asymptotic density of the even integers
(in the integers) is one half.

And a 1 gram ice cube will float in 1 gram of liquid water, but they both have the same number of molecules.
Spreading things out doesn't change how many there are.
EFQ

Two identical spheres have twice as much volume
as one of these spheres.

[Peter Percival]

Post by peteolcott

Post by peteolcott

Post by peteolcott
I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

http://youtu.be/s86-Z-CbaHA
Maybe even you can understand it if you don't have to read.
EFQ

if Banach-Tarski asserts it decomposes a sphere into
parts and then recompose two spheres

each identical to the original

Not identical, rather of equal volume.

Post by peteolcott

Post by peteolcott
from these same
parts then I have proven that Banach-Tarski is asserting
a falsehood.
The first two minutes of the video confirm that my
understanding of the Banach-Tarski assertion is correct.
The details after this assertion are moot as long as
I prove that the assertion itself is impossible.
I take exactly this same approach on the 1931 GIT. I do
not need to understand any detail of the 1931 GIT to
utterly refute it as long as I prove that its conclusion
is impossible.
Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.
Copyright 2016, 2017, 2018 Pete Olcott

[peteolcott]

Post by Peter Percival

Post by peteolcott

Post by peteolcott

Post by peteolcott
I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

No you haven't.  Try watching this: http://youtu.be/s86-Z-CbaHA
Maybe even you can understand it if you don't have to read.
EFQ

if Banach-Tarski asserts it decomposes a sphere into
parts and then recompose two spheres

each identical to the original

Not identical, rather of equal volume.

If we were to do Banach-Tarski with physical matter
instead of geometric spheres it would be much more
obvious that a mistake has been made because each
ball would have half the mass of the original ball.

Post by Peter Percival

Post by peteolcott

Post by peteolcott
from these same
parts then I have proven that Banach-Tarski is asserting
a falsehood.
The first two minutes of the video confirm that my
understanding of the Banach-Tarski assertion is correct.
The details after this assertion are moot as long as
I prove that the assertion itself is impossible.
I take exactly this same approach on the 1931 GIT. I do
not need to understand any detail of the 1931 GIT to
utterly refute it as long as I prove that its conclusion
is impossible.
Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.
Copyright 2016, 2017, 2018 Pete Olcott

[Peter Percival]

Post by peteolcott

Post by Peter Percival

Post by peteolcott

Post by peteolcott

Post by peteolcott
I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

http://youtu.be/s86-Z-CbaHA
Maybe even you can understand it if you don't have to read.
EFQ

if Banach-Tarski asserts it decomposes a sphere into
parts and then recompose two spheres

each identical to the original

Not identical, rather of equal volume.

If we were to do Banach-Tarski with physical matter

Which we can't. You might as well begin 'if black were white...' Is
that the problem, that you think that the Banach–Tarski paradox is about
physical matter?

Post by peteolcott
instead of geometric spheres it would be much more
obvious that a mistake has been made because each
ball would have half the mass of the original ball.

[peteolcott]

Post by peteolcott
If we were to do Banach-Tarski with physical matter

Then you aren't doing Banach-Tarski.
Banach-Tarski says nothing at all about physical matter. You keep trying to refute something by redefining the original terms. That's not valid.

Yuz too dum to have ever herd of an anal logy?

All analogies are imperfect.

Analogies are always perfect along at least one dimension of comparison.

And what about all the other dimensions? The only perfect representation of something is the thing itself. Everything else is imperfect.

Yes, but, you but you screwed up by saying that the
analogy itself was not perfect. With this degree of
lack of academic discipline you will never understand me.

The analogy itself IS either a perfect analogy along
at least one dimension or not an analogy at all.

That's why they're analogies, rather than the actual thing.

EFQ

[Exfalso Quodlibet]

Post by peteolcott

Post by peteolcott
If we were to do Banach-Tarski with physical matter

Then you aren't doing Banach-Tarski.
Banach-Tarski says nothing at all about physical matter. You keep trying to refute something by redefining the original terms. That's not valid.

Yuz too dum to have ever herd of an anal logy?

All analogies are imperfect.

Analogies are always perfect along at least one dimension of comparison.

And what about all the other dimensions? The only perfect representation of something is the thing itself. Everything else is imperfect.

Yes, but, you but you screwed up by saying that the
analogy itself was not perfect.

Your analogy was imperfect.

Post by peteolcott
The analogy itself IS either a perfect analogy along
at least one dimension or not an analogy at all.

Then your analogy was no analogy. Banach-Tarski cannot be analogized to physical matter, because physical balls do not have an infinite number of ball particles to work with.

EFQ

[Peter Percival]

Post by peteolcott
I take exactly this same approach on the 1931 GIT. I do
not need to understand any detail of the 1931 GIT to

Cranks have no sense of shame, do they?

Post by peteolcott
utterly refute it as long as I prove that its conclusion
is impossible.
Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.
Copyright 2016, 2017, 2018 Pete Olcott

[peteolcott]

Post by Peter Percival

Post by peteolcott
I take exactly this same approach on the 1931 GIT. I do
not need to understand any detail of the 1931 GIT to

Cranks have no sense of shame, do they?

Post by peteolcott
utterly refute it as long as I prove that its conclusion
is impossible.

This truism may be over your head:
As long as any conclusion has been shown to be impossible
the reasoning leading to this impossible conclusion is
proven to be necessarily incorrect without even looking at it.

Are you aware that personal insults are not any correct
form of rebuttal? Some otherwise intelligent people seem
to be quite stupid on this key point.

Post by Peter Percival

Post by peteolcott
Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.
Copyright 2016, 2017, 2018 Pete Olcott

[Peter Percival]

Post by peteolcott

Post by Peter Percival

Post by peteolcott
I take exactly this same approach on the 1931 GIT. I do
not need to understand any detail of the 1931 GIT to

Cranks have no sense of shame, do they?

Post by peteolcott
utterly refute it as long as I prove that its conclusion
is impossible.

As long as any conclusion has been shown to be impossible
the reasoning leading to this impossible conclusion is
proven to be necessarily incorrect without even looking at it.
Are you aware that personal insults are not any correct
form of rebuttal? Some otherwise intelligent people seem
to be quite stupid on this key point.

I am aware that some people are so impervious to logical argumentation
that some people may well think that personal insults are all that's
left. But so what? I asked 'Cranks have no sense of shame, do they?'.
What's the answer?

Post by peteolcott

Post by Peter Percival

Post by peteolcott
Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.

Just suppose that the Banach–Tarski paradox was called the
Banach–Tarski theorem. Would you have felt the need to comment on it?

[Mike Terry]

On 22/08/2018 17:16, Peter Percival wrote:
...

Post by Peter Percival
I am aware that some people are so impervious to logical argumentation
that some people may well think that personal insults are all that's
left. But so what? I asked 'Cranks have no sense of shame, do they?'.
What's the answer?

I think cranks, and PO specifically probably do have a sense of shame,
although clearly they are not ashamed to pronounce confidently on
subjects of which they have almost no understanding.

The problem here is that PO is deluded regarding, well, so many things!,
that he genuinely doesn't perceive his lack of understanding/competence,
particularly in comparison with others. On the contrary - he sees
himself as mentally far superior to other people he's required to
interact with. So even though he may concede that he doesn't
necessarily follow all the technical details of a subject, he none the
less has a far better _insight_ into the subject than others, since he
is able to "focus intensely" on the core essence of the subject and so
to identify where others are going astray!

So he sees no reason to be ashamed in this situation - he probably
believes he's ultimately helping humanity etc.. However, I expect there
could be all sorts of other situations (not relating to his delusions)
where he would still feel shame like anyone else.


Mike.

[peteolcott]

...

Post by Peter Percival
I am aware that some people are so impervious to logical argumentation
that some people may well think that personal insults are all that's
left.  But so what?  I asked 'Cranks have no sense of shame, do they?'.
What's the answer?

I think cranks, and PO specifically probably do have a sense of shame, although clearly they are not ashamed to pronounce confidently on subjects of which they have almost no understanding.
The problem here is that PO is deluded regarding, well, so many things!, that he genuinely doesn't perceive his lack of understanding/competence, particularly in comparison with others.  On the contrary - he sees himself as mentally far superior to other
people he's required to interact with.  So even though he may concede that he doesn't necessarily follow all the technical details of a subject, he none the less has a far better _insight_ into the subject than others, since he is able to "focus intensely"
on the core essence of the subject and so to identify where others are going astray!
So he sees no reason to be ashamed in this situation - he probably believes he's ultimately helping humanity etc..  However, I expect there could be all sorts of other situations (not relating to his delusions) where he would still feel shame like anyone
else.
Mike.

You are quite goofy. Does that feel good?
If you can point out any actual mistakes that I have
made feel free, otherwise realize that you are goofy.

[Peter Percival]

You [Mike Terr] are quite goofy. Does that feel good?
If you can point out any actual mistakes that I have
made

When you "refute" the Banach–Tarski paradox by claiming that the parts
into which the sphere is decomposed are themselves spheres and that it
is about material spheres, these mistakes are pointed out to you. But
there is a bigger mistake that runs through all your post: your belief
that not learning anything about your subject us a virtue. It isn't.

feel free, otherwise realize that you are goofy.

[peteolcott]

You [Mike Terr] are quite goofy. Does that feel good?
If you can point out any actual mistakes that I have
made

When you "refute" the Banach–Tarski paradox by claiming that the parts into which the sphere is decomposed are themselves spheres

I NEVER said anything like this, that would be a very stupid thing to say.

and that it is about material spheres, these mistakes are pointed out to you.  But there is a bigger mistake that runs through
all your post: your belief that not learning anything about your subject us a virtue.  It isn't.

feel free, otherwise realize that you are goofy.

[peteolcott]

Post by peteolcott
Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.

I know you think that every word should have just one meaning,

There exists a set of unique semantic meanings.
That very many of these meanings are tied to the same word
greatly hinders the effectiveness of the communication process.

Any formalized system of this unique set of meanings
would have a unique integer value for each meaning.

but "paradox" has more than one.  One of those meanings is "contrary to the man in the street's expectations"; and it is that kind of paradox that Banach–Tarski's is.  But the man in the street
(like the Olcott in his box) cannot be expected to be familiar with the farther reaches of mathematics.

Here is my first draft of a definition:

A paradox is the application of what seems to be correct
reasoning to any set of seemingly true premises such that
a contradiction is derived.

Copyright 2018 Pete Olcott

// Olcott's Truth schema
∀L ∈ Formal_Systems True(L, C) ↔ ∃Γ ⊆ Axioms(L) (Γ ⊢ C)

// Olcott's Truth predicate for L
True(C) ↔ ∃Γ ⊆ Axioms(L) (Γ ⊢ C)

// Formalized Liar Paradox for L
LP ↔ ~∃Γ ⊆ Axioms(L) (Γ ⊢ LP) // ~(LP ∨ ~LP)

<begin quoted material>
Page 254 Chapter 10 The Relation of Prolog to Logic
Programming in Prolog Using the ISO Standard Fifth Edition by Clocksin and Mellish
?- equal(foo(Y), Y).

...match a term against an uninstantiated subterm of itself...
...So Y ends up standing for some kind of infinite structure...
<end quoted material>

?- ↔(True(LP), LP).
So LP ends up standing for some kind of infinite structure.

Copyright 2016, 2017, 2018 Pete Olcott

[Peter Percival]

Post by peteolcott

Post by peteolcott
Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.

I know you think that every word should have just one meaning,

There exists a set of unique semantic meanings.

No, there just doesn't, and the proof is immediate. Look in a
dictionary, a lot (most?) of the words in it have more than one meaning.

Post by peteolcott
That very many of these meanings are tied to the same word
greatly hinders the effectiveness of the communication process.
Any formalized system of this unique set of meanings
would have a unique integer value for each meaning.

but "paradox" has more than one.  One of those meanings is "contrary
to the man in the street's expectations"; and it is that kind of
paradox that Banach–Tarski's is.  But the man in the street (like the
Olcott in his box) cannot be expected to be familiar with the farther
reaches of mathematics.

There is no need for you to draft a definition, we can all look it up in
a dictionary if we have any doubts about it. If your definition is one
among the dictionary definitions, then you are wasting your breath, if
it isn't, then you are wrong.

Post by peteolcott
A paradox is the application of what seems to be correct
reasoning to any set of seemingly true premises such that
a contradiction is derived.
Copyright 2018 Pete Olcott
// Olcott's Truth schema
∀L ∈ Formal_Systems True(L, C) ↔ ∃Γ ⊆ Axioms(L) (Γ ⊢ C)
// Olcott's Truth predicate for L
True(C) ↔ ∃Γ ⊆ Axioms(L) (Γ ⊢ C)
// Formalized Liar Paradox for L
LP ↔ ~∃Γ ⊆ Axioms(L) (Γ ⊢ LP)  // ~(LP ∨ ~LP)
<begin quoted material>
Page 254 Chapter 10 The Relation of Prolog to Logic
Programming in Prolog Using the ISO Standard Fifth Edition by Clocksin and Mellish
?- equal(foo(Y), Y).
...match a term against an uninstantiated subterm of itself...
...So Y ends up standing for some kind of infinite structure...
<end quoted material>
?- ↔(True(LP), LP).
So LP ends up standing for some kind of infinite structure.
Copyright 2016, 2017, 2018 Pete Olcott

[peteolcott]

Post by peteolcott

Post by peteolcott
Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.

I know you think that every word should have just one meaning,

There exists a set of unique semantic meanings.

No, there just doesn't, and the proof is immediate.  Look in a dictionary, a lot (most?) of the words in it have more than one meaning.

Post by peteolcott
That very many of these meanings are tied to the same word
greatly hinders the effectiveness of the communication process.
Any formalized system of this unique set of meanings
would have a unique integer value for each meaning.

but "paradox" has more than one.  One of those meanings is "contrary to the man in the street's expectations"; and it is that kind of paradox that Banach–Tarski's is.  But the man in the street (like the Olcott in his box) cannot be expected to be
familiar with the farther reaches of mathematics.

There is no need for you to draft a definition, we can all look it up in a dictionary if we have any doubts about it.  If your definition is one among the dictionary definitions, then you are wasting your breath, if it isn't, then you are wrong.

Many of the things you say are too stupid to respond to.
I am going you this feedback now so that you will know
exactly why I will not be responding to any poor quality replies.

They are not stupid because you have a low IQ or don't know better.
I would be rude of me to call you stupid for things that you cannot control.

You are very intelligent and have pretty good knowledge,
yet as in this case say very stupid things by not bothering
to think before you speak.

Post by peteolcott
A paradox is the application of what seems to be correct
reasoning to any set of seemingly true premises such that
a contradiction is derived.

EVERY OTHER DEFINITION IS TOO VAGUE.

Post by peteolcott
Copyright 2018 Pete Olcott
// Olcott's Truth schema
∀L ∈ Formal_Systems True(L, C) ↔ ∃Γ ⊆ Axioms(L) (Γ ⊢ C)
// Olcott's Truth predicate for L
True(C) ↔ ∃Γ ⊆ Axioms(L) (Γ ⊢ C)
// Formalized Liar Paradox for L
LP ↔ ~∃Γ ⊆ Axioms(L) (Γ ⊢ LP)  // ~(LP ∨ ~LP)
<begin quoted material>
Page 254 Chapter 10 The Relation of Prolog to Logic
Programming in Prolog Using the ISO Standard Fifth Edition by Clocksin and Mellish
?- equal(foo(Y), Y).
...match a term against an uninstantiated subterm of itself...
...So Y ends up standing for some kind of infinite structure...
<end quoted material>
?- ↔(True(LP), LP).
So LP ends up standing for some kind of infinite structure.
Copyright 2016, 2017, 2018 Pete Olcott

[peteolcott]

Post by peteolcott

Post by peteolcott
Olcott's KEY hypothesis (to be progressively proven)
Every paradox has a hidden error somewhere.

I know you think that every word should have just one meaning,

There exists a set of unique semantic meanings.

Prove it. I'm sure the semanticists down the hall in the linguistics program would love to have access to that set!

It is self evident (to me) that there is a unique and finite set of currently existing atomic semantic meanings.

No one can ever prove the self evident, (AKA tautology) all that can be done is to explain it in progressively clearer detail.

https://www.britannica.com/topic/tautology
Tautology, in logic, a statement so framed that it cannot be denied
without inconsistency. Thus, “All humans are mammals” is held to
assert with regard to anything whatsoever that either it is a human
or it is not a mammal. But that universal “truth” follows not from
any facts noted about real humans but only from the actual use of
human and mammal and is thus purely a matter of definition.

17th century philosophers like Douet, Lodwick, Urquhart, Dalgarno, and Wilkins thought the same way, but that was pretty much a fad that died out in the 18th century, after it proved to be unworkable. It still rises up every now and again (e.g. Weilgart's aUI language), but it's mostly just cranks who do it these days.

Post by peteolcott
That very many of these meanings are tied to the same word
greatly hinders the effectiveness of the communication process.

In fact, communication evolved for efficiency, and part of efficiency is homophony. We're really good at using context to disambiguate meaning.

Post by peteolcott
Any formalized system of this unique set of meanings
would have a unique integer value for each meaning.

So you can only have a countably infinite number of meanings? Goodbye real number line!
EFQ

addressed above.

[DKleinecke]

Post by peteolcott
It is self evident (to me) that there is a unique and
finite set of currently existing atomic semantic meanings.

It is far from self-evident to the rest of us.

But it might be an acceptable hypothesis were we able to
understand what it means.

"Unique" seems to add nothing. All sets are unique. I
cannot assign any content to "currently existing".

But what is more important is what does "meaning" mean
in this sentence?

[peteolcott]

Post by DKleinecke

Post by peteolcott
It is self evident (to me) that there is a unique and
finite set of currently existing atomic semantic meanings.

It is far from self-evident to the rest of us.
But it might be an acceptable hypothesis were we able to
understand what it means.
"Unique" seems to add nothing. All sets are unique. I
cannot assign any content to "currently existing".
But what is more important is what does "meaning" mean
in this sentence?

Each of the sometimes extremely subtle nuances
of meaning must be distinguished from each another
rather than conflated together as it typically the
case.

They must be specified in an inheritance hierarchy
knowledge ontology where no semantic meaning is ever
allowed to override any of the meanings further up
the hierarchy. https://en.wikipedia.org/wiki/Ontology_(information_science)
All of the specified semantic meaning must be additive
to the meaning already specified further up in the
hierarchy.

// As soon as this is sufficiently specified the
// notion of anything totally containing itself
// will be understood to be impossible, thus eliminating
// Russell's Paradox before it gets started.
total containment
conceptual total containment
physical total containment

There is one more key architectural criteria (besides
zero overrides) that must be kept as a trade secret.

The material above along with the trade secret material
fully specifies the fundamental architecture of the natural
preexisting order of the collection of all knowledge.

Copyright 2016, 2017, 2018 Pete Olcott

[Peter Percival]

Post by peteolcott

Post by DKleinecke

Post by peteolcott
It is self evident (to me) that there is a unique and
finite set of currently existing atomic semantic meanings.

It is far from self-evident to the rest of us.
But it might be an acceptable hypothesis were we able to
understand what it means.
"Unique" seems to add nothing. All sets are unique. I
cannot assign any content to "currently existing".
But what is more important is what does "meaning" mean
in this sentence?

Each of the sometimes extremely subtle nuances
of meaning must be distinguished from each another
rather than conflated together as it typically the
case.
They must be specified in an inheritance hierarchy
knowledge ontology where no semantic meaning is ever
allowed to override any of the meanings further up
the hierarchy. https://en.wikipedia.org/wiki/Ontology_(information_science)
All of the specified semantic meaning must be additive
to the meaning already specified further up in the
hierarchy.
// As soon as this is sufficiently specified the
// notion of anything totally containing itself
// will be understood to be impossible, thus eliminating
// Russell's Paradox before it gets started.
total containment
  conceptual total containment
  physical total containment
There is one more key architectural criteria (besides
zero overrides) that must be kept as a trade secret.

I.e., The Olcott doesn't know what it is.

Post by peteolcott
The material above along with the trade secret material
fully specifies the fundamental architecture of the natural
preexisting order of the collection of all knowledge.
Copyright 2016, 2017, 2018 Pete Olcott

[peteolcott]

Post by Peter Percival

Post by peteolcott

Post by DKleinecke

Post by peteolcott
It is self evident (to me) that there is a unique and
finite set of currently existing atomic semantic meanings.

It is far from self-evident to the rest of us.
But it might be an acceptable hypothesis were we able to
understand what it means.
"Unique" seems to add nothing. All sets are unique. I
cannot assign any content to "currently existing".
But what is more important is what does "meaning" mean
in this sentence?

Each of the sometimes extremely subtle nuances
of meaning must be distinguished from each another
rather than conflated together as it typically the
case.
They must be specified in an inheritance hierarchy
knowledge ontology where no semantic meaning is ever
allowed to override any of the meanings further up
the hierarchy. https://en.wikipedia.org/wiki/Ontology_(information_science)
All of the specified semantic meaning must be additive
to the meaning already specified further up in the
hierarchy.
// As soon as this is sufficiently specified the
// notion of anything totally containing itself
// will be understood to be impossible, thus eliminating
// Russell's Paradox before it gets started.
total containment
   conceptual total containment
   physical total containment
There is one more key architectural criteria (besides
zero overrides) that must be kept as a trade secret.

I.e., The Olcott doesn't know what it is.

I have taken to only glancing at one or two of your
posts and not seeing anything of substance skipping
all the rest.

One flippant comment gets me to ignore all of your comments
for that whole day. With other people I am a little less
harsh. For them I just ignore their flippant comments. I
treat you differently because you have many more comments
and I need to avoid wasting time.

Post by Peter Percival

Post by peteolcott
The material above along with the trade secret material
fully specifies the fundamental architecture of the natural
preexisting order of the collection of all knowledge.
Copyright 2016, 2017, 2018 Pete Olcott

[Peter Percival]

Post by peteolcott
I have taken to only glancing at one or two of your
posts and not seeing anything of substance skipping
all the rest.

I can assure you (should you need assuring) that ignoring any, or even
all, of my posts will not have a negative effect on your knowledge or
what you might learn. That is because you think that ignorance and not
learning is a virtue. The Banach–Tarski matter is a new thing, but for
_years_ you have commented on logical matters without bothering to learn
anything about logic. For example, I recall you being unable to draw a
truth-table of formulae with negation signs in them. You claim to have
taken all the courses necessary to get a degree in computer science, but
you know nothing about Turing machines, compression algorithms and SAT.

If you wish to discuss the Banach–Tarski paradox you will need to learn
some measure theory and some group theory. Will you? No. Actually,
you are right to not even start to learn about such things. Both topics
will be over your head and you will waste your time when you could be
earning your keep as a waster-upper in MacDonalds. You don't even know
what a (mathematical) sphere is, do you?

Isn't there a tall building, or a bridge, or something, that you could
throw yourself off?

[Peter Percival]

Post by peteolcott
No one can ever prove the self evident, (AKA tautology

Your claim that tautologies can't be proved is contradicted by the
Britannica quotation.

Post by peteolcott
) all that can be
done is to explain it in progressively clearer detail.
    https://www.britannica.com/topic/tautology
    Tautology, in logic, a statement so framed that it cannot be denied
    without inconsistency

In other words, one can prove that something is a tautology by showing
that its negation is inconsistent. (There ate other ways, too.)

Post by peteolcott
. Thus, “All humans are mammals” is held to
    assert with regard to anything whatsoever that either it is a human
    or it is not a mammal. But that universal “truth” follows not from
    any facts noted about real humans but only from the actual use of
    human and mammal and is thus purely a matter of definition.

[peteolcott]

Post by peteolcott
I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

No you haven't. Try watching this: http://youtu.be/s86-Z-CbaHA
Maybe even you can understand it if you don't have to read.
EFQ

He has an enormously convoluted process for naming the points.
A much simpler process would be to use a Cartesian space with
(0,0,0) as the center of a one inch sphere. We can add
georeferencing by simply placing the center of the sphere at
the center of the Earth.

Now we have a set of points unique in the universe that
can each be easily referenced by an x,y,z triple.

It seems to me that the confusion of Banach-Tarski is
that the notion of infinity has never been specified
correctly.

For example we could remove the set of points specifying
a circle from the sphere and confuse ourselves into
believing that there is no gap what-so-ever from this
missing piece of the sphere.

We could then remove enough of these circles to have what
seems to be two spheres, never noticing that even an
infinitesimal gap and the original sphere ceases to be spherical.

Copyright 2016, 2017, 2018 Pete Olcott

[Peter Percival]

Post by peteolcott
It seems to me that the confusion of Banach-Tarski is
that the notion of infinity has never been specified
correctly.

What is the correct specification of the notion of infinity?

[peteolcott]

Post by peteolcott

Post by peteolcott
I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

No you haven't. Try watching this: http://youtu.be/s86-Z-CbaHA
Maybe even you can understand it if you don't have to read.

He has an enormously convoluted process for naming the points.

If you think combinations of up, down, left, and right are "enormously convoluted", you're stupider than I thought.

Post by peteolcott
It seems to me that the confusion of Banach-Tarski is
that the notion of infinity has never been specified
correctly.

No one is confused but you, and the specification is there, but you find it "enormously convoluted".

They are much more convoluted than a point in Cartesian space.

Post by peteolcott
For example we could remove the set of points specifying
a circle from the sphere and confuse ourselves into
believing that there is no gap what-so-ever from this
missing piece of the sphere.

There is no gap in the volume, the surface area, or the cardinality of the set of points.

This infinitesimal gap causes the object to cease to meet the specification of a sphere.

If you take out all of the even integers from the set of all integers, you still have as many integers as you started with,

Yet no longer the complete set of integers, some are missing. Likewise with the sphere having a circle missing.

Copyright 2018 Pete Olcott

but if you take out all of the integers from the set of all integers, you are left with nothing.
When you subtract one infinity from another, the result depends on which precise sets you are dealing with. Specifying those sets requires using methods that are apparently too "enormously convoluted" for your peabrain to comprehend.
EFQ

[Jim Burns]

Post by peteolcott

On Tuesday, August 21, 2018 at 10:18:55 PM UTC-4,

Post by peteolcott
I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

http://youtu.be/s86-Z-CbaHA
Maybe even you can understand it if you don't have to read.

He has an enormously convoluted process for naming the points.

This seems to be your standard objection.
I suppose that your use of this objection means that
you have none stronger -- which is what I would expect,
but it would be nice if somehow you recognized that, too.

Post by peteolcott
A much simpler process would be to use a Cartesian space with
(0,0,0) as the center of a one inch sphere. We can add
georeferencing by simply placing the center of the sphere at
the center of the Earth.
Now we have a set of points unique in the universe that
can each be easily referenced by an x,y,z triple.

Some of these points are members of a set S which obeys the
definition of a sphere. Other points are members of a
different set S' which also obeys the definition of a set.
We say that the sets S and S' _are spheres_ .

If there is some function f that accepts points u and v in
sphere S returns points u' and v' in sphere S' _and the_
_distance is preserved_
|u - v| = |f(u) - f(v)| = |u' - v'|
then we say that f is a _rigid translation_ of the points
in S to points in S'.

Banach-Tarski is a proof that there are subsets A,B,C,D,E
of S which jointly hold all of S but have no point twice
(they partition S) and for which there are _rigid_
_translations_ f_A, f_B, f_C, f_D, f_E to either of spheres
S' or S" -- and which cover all of the points in S' and S".

Please don't confuse what I've written here with a proof.
I haven't proven anything, I've just tried to make clear
what is being discussed. If you counter my claim here with
your own claim -- that there _aren't_ such f_A, f_B, f_C,
f_D, f_E -- you haven't addressed the proof, which is
elsewhere.

Post by peteolcott
It seems to me that the confusion of Banach-Tarski is
that the notion of infinity has never been specified
correctly.

It seems to me that the notions of length, area, volume,
hypervolume, ... only can be correctly applied to some
_but not all_ sets of points. For a non-measurable set,
it is as though we ask "What is the sum of 5 and purple?"

(I had thought you were familiar with that concept, but
I guess you are currently choosing ignorance.)

What we want is for the pieces of a sphere, broken apart
and re-assembled, to preserve volume. That doesn't happen
with Banach-Tarski because the pieces into which S is
broken cannot correctly be said to have volumes. This shines
a light onto what we mean by volume, area, etc, but it is
not an error.

The error (which, by your principle, must exist) was in
thinking that our uninformed intuitions about volume,
area, etc, were perfect.

Post by peteolcott
For example we could remove the set of points specifying
a circle from the sphere and confuse ourselves into
believing that there is no gap what-so-ever from this
missing piece of the sphere.
We could then remove enough of these circles to have what
seems to be two spheres, never noticing that even an
infinitesimal gap and the original sphere ceases to be
spherical.

You're presenting some sort of hypothetical whereby,
if we did such and such and didn't notice it, we would
be making an error and not noticing it. You don't
address the Banach-Tarski theorem, you imagine addressing
it, and -- in your imagination -- you vanquish it.

Other than making you feel good (I assume), this is pointless.
Go. Read.

[peteolcott]

Post by Jim Burns

Post by peteolcott

On Tuesday, August 21, 2018 at 10:18:55 PM UTC-4,

Post by peteolcott
I have proven that the sphere has been duplicated
by copying its points to another sphere, thus
refuting Banach-Tarski.

No you haven't.  Try watching this: http://youtu.be/s86-Z-CbaHA
Maybe even you can understand it if you don't have to read.

He has an enormously convoluted process for naming the points.

This seems to be your standard objection.
I suppose that your use of this objection means that
you have none stronger -- which is what I would expect,
but it would be nice if somehow you recognized that, too.

Post by peteolcott
A much simpler process would be to use a Cartesian space with
(0,0,0) as the center of a one inch sphere. We can add
georeferencing by simply placing the center of the sphere at
the center of the Earth.
Now we have a set of points unique in the universe that
can each be easily referenced by an x,y,z triple.

Some of these points are members of a set S which obeys the
definition of a sphere. Other points are members of a
different set S' which also obeys the definition of a set.
We say that the sets S and S' _are spheres_ .
If there is some function f that accepts points u and v in
sphere S returns points u' and v' in sphere S' _and the_
_distance is preserved_
 |u - v|  =  |f(u) - f(v)|  =  |u' - v'|
then we say that f is a _rigid translation_ of the points
in S to points in S'.
Banach-Tarski is a proof that there are subsets A,B,C,D,E
of S which jointly hold all of S but have no point twice
(they partition S) and for which there are _rigid_
_translations_ f_A, f_B, f_C, f_D, f_E to either of spheres
S' or S" -- and which cover all of the points in S' and S".
Please don't confuse what I've written here with a proof.
I haven't proven anything, I've just tried to make clear
what is being discussed. If you counter my claim here with
your own claim -- that there _aren't_ such f_A, f_B, f_C,
f_D, f_E -- you haven't addressed the proof, which is
elsewhere.

Post by peteolcott
It seems to me that the confusion of Banach-Tarski is
that the notion of infinity has never been specified
correctly.

It seems to me that the notions of length, area, volume,
hypervolume, ... only can be correctly applied to some
_but not all_ sets of points. For a non-measurable set,
it is as though we ask "What is the sum of 5 and purple?"
 (I had thought you were familiar with that concept, but
 I guess you are currently choosing ignorance.)
What we want is for the pieces of a sphere, broken apart
and re-assembled, to preserve volume. That doesn't happen
with Banach-Tarski because the pieces into which S is
broken cannot correctly be said to have volumes. This shines
a light onto what we mean by volume, area, etc, but it is
not an error.
The error (which, by your principle, must exist) was in
thinking that our uninformed intuitions about volume,
area, etc, were perfect.

Post by peteolcott
For example we could remove the set of points specifying
a circle from the sphere and confuse ourselves into
believing that there is no gap what-so-ever from this
missing piece of the sphere.
We could then remove enough of these circles to have what
seems to be two spheres, never noticing that even an
infinitesimal gap and the original sphere ceases to be
spherical.

You're presenting some sort of hypothetical whereby,
if we did such and such and didn't notice it, we would
be making an error and not noticing it. You don't
address the Banach-Tarski theorem, you imagine addressing
it, and -- in your imagination -- you vanquish it.
Other than making you feel good (I assume), this is pointless.
Go. Read.

The provided example video says that there are gaps in the spheres.
http://youtu.be/s86-Z-CbaHA

The first big mistake is when he names the same point twice with noticing that he did this:
RUL PURPLE **
DD BLUE
RDR RED
U ORANGE **
Now when he moves RUL and does not move U he has just copied a point without knowing it.

[Jim Burns]

Post by peteolcott

Post by Jim Burns
You're presenting some sort of hypothetical whereby,
if we did such and such and didn't notice it, we would
be making an error and not noticing it. You don't
address the Banach-Tarski theorem, you imagine addressing
it, and -- in your imagination -- you vanquish it.
Other than making you feel good (I assume),
this is pointless. Go. Read.

The provided example video says that there are gaps
in the spheres.
http://youtu.be/s86-Z-CbaHA

Q: "What's an anagram of Banach-Tarski?"
A: "Banach-Tarski Banach-Tarski."

Post by peteolcott
The first big mistake is when he names the same point twice
RUL PURPLE **
DD  BLUE
RDR RED
U ORANGE **
Now when he moves RUL and does not move U he has just
copied a point without knowing it.

Okay, I'm going to accept this as an attempt on your part
to make some sort of argument. I have pretty much accused
you of not even knowing what an argument is, but this is
an argument. I withdraw that claim.

It's a wrong argument, but it gives me more to work with
than just you repeating a claim (something that you seemed
to be addicted to doing).

I encourage you to make more _arguments_ , even wrong
arguments. TIA.

----
This objection of yours is wrong _because_ the point RUL
(ending in R, so purple)
(arrived at by moving, from the origin, left-up-right on
the sphere's surface)
is not the same point as the point U (ending in U, so orange)
(arrived at by moving up)

You're moving on the surface _of a sphere_ .
On a plane, left-up-right RUL would land you at the same point
as up U, but this is not a plane, it's a sphere.

Yes, for small enough steps right-left-up-down RLUD,
the endpoints of RUL and U will be "close enough" to
the same point for any positive ammount of error in
what "close enough" is. (We could say "In the limit,
the surface of a sphere is flat.")

However, in this video, we are given a fixed size step,
cos(1/2). We are NOT allowed to shrink our step size down
until RUL is close enough (whatever that might be) to U.

----
The point RUL is not the point U.
I think the easiest way to see this is to _increase_ the
size of the steps RLUD until four of them take us all
the way around the globe and back home again.

(These are not the same as the Banach-Tarski steps.
For the Banach-Tarski steps, we don't want to return
home, not for any finite number of steps. I only intend
to show that traveling on a sphere and traveling on a plane
are different.)

You start at Home, point H, facing north.
Ahead of you is the Up pole, behind you is the Down pole.
To your left, the Left Pole, to your right, the Right pole.

Step to your left.
You are at the Left pole, facing north.
Ahead of you is the Up pole, to your right is Home.

Step north.
You are at the Up pole, with the left pole behind you.
To your right is Home.

Step right.
You are Home.

So, with the quarter-globe steps, RUL brings you back Home.
But U takes you to the Up pole.
These are not the same.

----
The Banach-Tarski paradox does depend upon the concept
of infinity, as you said. That you don't want Banach-
-Tarski to be valid does not make the concept of
infinity wrong somehow.

----
We can make the same sort of argument for the natural
numbers.
(Though it's not as impressive as the Banach-Tarski
result. Spheres have a more material-like appearance,
for example, like chocolate, as the video points out.)

Take all the natural numbers and label them with their
binary numerals: 0, 1, 10, 11, 100, 101, 110, 111, 1000, ...

Take all those labels and append a '0' to their right end:
00, 10, 100, 110, 1000, 1010, 1100, ...
These are the label for the even numbers.

Now, take all those original labels and append a '1':
01, 11, 101, 111, 1001, 1011, 1101, ...
These are the labels for the odd numbers.

We can take the even numbers and map them to all
the numbers, and we can take the odd numbers and map
them to all the numbers. In a sense, there are
two copies of the natural numbers inside the natural
numbers.

This is not some sort of error. It's only what it
means to be infinite.

----
Consider a one-ended chain. You can hold that unique link
at which it ends in your hand, but there is no other
link that is an end. Follow the chain across fields,
across the oceans, wrapped around the world, stretched
to the moon, to the stars, to the end of the universe
and beyond: no link is another end.

This chain is literally infinite: not-ended on one side.
Yes, it's imaginary, but we can describe it. Every link
except the unique end in your hand is directly connect
to exactly two other links.
(We can add more description to make sure there are
no _extras_ but this is enough for _at least_ one
infinite chain.)

Obviously, this is not at all a normal chain. And it
has some highly abnormal properties. For example, if
one deletes every other link, joining the two to keep
the chain continuous, the other end of a _normal_
chain would be half as far away. _But this chain doesn't_
_have "the other" end_ . If one deletes every other link
of a one-ended chain, what one is left with is still
a one-ended chain.

And so on.

Why would a one-ended chain be like a two-ended chain,
anyway?

[peteolcott]

Post by Jim Burns

Post by peteolcott

Post by Jim Burns
You're presenting some sort of hypothetical whereby,
if we did such and such and didn't notice it, we would
be making an error and not noticing it. You don't
address the Banach-Tarski theorem, you imagine addressing
it, and -- in your imagination -- you vanquish it.
Other than making you feel good (I assume),
this is pointless. Go. Read.

The provided example video says that there are gaps
in the spheres.
http://youtu.be/s86-Z-CbaHA

Q: "What's an anagram of Banach-Tarski?"
A: "Banach-Tarski Banach-Tarski."

Post by peteolcott
The first big mistake is when he names the same point twice
RUL PURPLE **
DD  BLUE
RDR RED
U ORANGE **
Now when he moves RUL and does not move U he has just
copied a point without knowing it.

Okay, I'm going to accept this as an attempt on your part
to make some sort of argument. I have pretty much accused
you of not even knowing what an argument is, but this is
an argument. I withdraw that claim.

Here is a much better argument:
http://youtu.be/s86-Z-CbaHA

At 15:00 he defines and removes six subsets of points.
and then later on takes 1/6 of the points of the sphere
and is able to add many points to this sphere by merely
rotating it and nothing else so that it now has 75% of
the original points.

All of this is merely an artifact of his convoluted
naming convention. He simply confuses himself into
believing that actual points pop into existence by
magic, simply by rotating the sphere.

If we derive another sphere with a simpler naming convention.**
The point named by the Cartesian space xyz = (0,1,0) is
removed leaving a subset of the original sphere. There is
no possible rotation that ever puts this missing point back.

This unequivocally proves that the video version of
Banach–Tarski paradox is erroneous.

** A sphere with a one inch radius is geolocated such that
the center of this sphere is at the center of the Earth.
The poles of this sphere are defined by the line intersecting
the center of the poles of The Earth. {x,y,z} point {0,1,0}
is the north pole of this sphere.

Copyright Pete Olcott 2018.

[Jim Burns]

Post by Jim Burns
Okay, I'm going to accept this as an attempt on your part
to make some sort of argument. I have pretty much accused
you of not even knowing what an argument is, but this is
an argument. I withdraw that claim.

http://youtu.be/s86-Z-CbaHA
At 15:00 he defines and removes six subsets of points.
and then later on takes 1/6 of the points of the sphere
and is able to add many points to this sphere by merely
rotating it and nothing else so that it now has 75% of
the original points.

No, this is wrong. He doesn't add any point by rotating
them. The ending-in-R points and the ending-in-anything-
-except-L points are the same points rotated.
(It's anything _except L_ because we don't allow RL...
so removing R can't leave L... .)

For each ending-in-R point, there is either an
ending-in-R, ending-in-U, or ending-in-D point.

This is where the infiniteness of the points come in.
_That is not a mistake_ . The video gives a nice example
of the hyperdictionary, a which lists all possible words
composed of finite strings of abcdefghijklmnopqrstuvwxyz.

Suppose we break the hyperdictionary into 26 volumes.
the A-volume, B-volumne, ... , and Z-volume. Every word
in the A-volume is the letter 'a' followed by some
finite string of abcdefghijklmnopqrstuvwxyz.
But that's the complete hyperdictionary, only with
'a's in front. Does putting an 'a' in front of each
word in the hyperdictionary reduce the number of words
by a factor of 1/26?

Infinite things can contain multiple distinct copies
of their wholes. This is an essential aspect of being
infinite.

If you argue from your common sense about _finite_
collections, you will be in error. Do you remember
your principle about how there must be an error in
a paradox? There's your error.

All of this is merely an artifact of his convoluted
naming convention. He simply confuses himself into
believing that actual points pop into existence by
magic, simply by rotating the sphere.
If we derive another sphere

Let's not do that.

Instead, let's understand the Banach-Tarski argument.

Start with:
For each natural number k, there is a different even
number 2*k. No numbers are "popping into existence" here.
We are matching a natural number to an even number.

For this sense of "just as many", there are
just as many even numbers as all the natural numbers.

This sense of "just as many" should be familiar to you
from real life. Consider setting a dinner table.
You'll have six people at dinner. You put enough
plates on the table so that, for each person, there is
a different plate. That's enough plates.
(To make sure there aren't too many plates, match a
different diner to each plate. If you can do both,
match a plate to a diner and match a diner to a plate,
then there are enough but not too many plates. This
is what we call having the same _cardinality_ .)

[peteolcott]

Post by Jim Burns

Post by Jim Burns
Okay, I'm going to accept this as an attempt on your part
to make some sort of argument. I have pretty much accused
you of not even knowing what an argument is, but this is
an argument. I withdraw that claim.

http://youtu.be/s86-Z-CbaHA
At 15:00 he defines and removes six subsets of points.
and then later on takes 1/6 of the points of the sphere
and is able to add many points to this sphere by merely
rotating it and nothing else so that it now has 75% of
the original points.

No, this is wrong. He doesn't add any point by rotating
them. The ending-in-R points and the ending-in-anything-
-except-L points are the same points rotated.
 (It's anything _except L_ because we don't allow RL...
 so removing R can't leave L... .)

TimePoint: 16:48-16:55
"We have turned less than a quarter of the
sphere into nearly three quarters just by rotating it."

When we try the same thing using a much simpler and
unequivocal naming convention, we find that no sphere
can possibly gain any points simply by rotating it.

We remove the north pole of any sphere with a radius
of one inch, Cartesian Space point: (0.0, 1.0, 0.0)
we can see that no possible rotation will ever fill
in this missing point.

TimePoint: 16:09-16:57
The gain of points in the video is an artifact of
the naming conventions. When we rotate the sphere
some of the original points take on new names, and

some of the new names refer to points that do not
actually exist within this subset of points.

Copyright 2018 Pete Olcott

[Peter Percival]

Post by peteolcott
If we derive another sphere with a simpler naming convention.**
The point named by the Cartesian space xyz = (0,1,0) is
removed leaving a subset of the original sphere. There is
no possible rotation that ever puts this missing point back.

That has nothing to do with the Banach–Tarski paradox.

Post by peteolcott
This unequivocally proves that the video version of
Banach–Tarski paradox is erroneous.

[peteolcott]

Post by Peter Percival

Post by peteolcott
If we derive another sphere with a simpler naming convention.**
The point named by the Cartesian space xyz = (0,1,0) is
removed leaving a subset of the original sphere. There is
no possible rotation that ever puts this missing point back.

That has nothing to do with the Banach–Tarski paradox.

http://youtu.be/s86-Z-CbaHA

TimePoint: 16:48-16:55
"We have turned less than a quarter of the
sphere into nearly three quarters just by rotating it."

When we try the same thing using a much simpler and
unequivocal naming convention, we find that no sphere
can possibly gain any points simply by rotating it.

We remove the north pole of any sphere with a radius
of one inch, Cartesian Space point: (0.0, 1.0, 0.0)
we can see that no possible rotation will ever fill
in this missing point.

TimePoint: 16:09-16:57
The gain of points in the video is an artifact of
the naming conventions. When we rotate the sphere
some of the original points take on new names, and

some of the new names refer to points that do not
actually exist within this subset of points.

Copyright 2018 Pete Olcott

[peteolcott]

Post by peteolcott
All of this is merely an artifact of his convoluted
naming convention.

"I don't understand it" is not a proof that the conclusion is wrong.

http://youtu.be/s86-Z-CbaHA

TimePoint: 16:48-16:55
"We have turned less than a quarter of the
sphere into nearly three quarters just by rotating it."

When we try the same thing using a much simpler and
unequivocal naming convention, we find that no sphere
can possibly gain any points simply by rotating it.

We remove the north pole of any sphere with a radius
of one inch, Cartesian Space point: (0.0, 1.0, 0.0)
we can see that no possible rotation will ever fill
in this missing point.

TimePoint: 16:09-16:57
The gain of points in the video is an artifact of
the naming conventions. When we rotate the sphere
some of the original points take on new names, and

some of the new names refer to points that do not
actually exist within this subset of points.

Copyright 2018 Pete Olcott

[Exfalso Quodlibet]

Post by peteolcott
All of this is merely an artifact of his convoluted
naming convention.

"I don't understand it" is not a proof that the conclusion is wrong.

http://youtu.be/s86-Z-CbaHA
TimePoint: 16:48-16:55
"We have turned less than a quarter of the
sphere into nearly three quarters just by rotating it."
When we try the same thing using a much simpler

"I don't understand it" is not a proof that the conclusion is wrong.

EFQ

[Shobe, Martin]

Post by peteolcott
All of this is merely an artifact of his convoluted
naming convention.

"I don't understand it" is not a proof that the conclusion is wrong.

http://youtu.be/s86-Z-CbaHA
TimePoint: 16:48-16:55
"We have turned less than a quarter of the
 sphere into nearly three quarters just by rotating it."
When we try the same thing using a much simpler and
unequivocal naming convention, we find that no sphere
can possibly gain any points simply by rotating it.

Like many things that you choose to pontificate about, you are woefully
ignorant about points in spheres. Every sphere, regardless of size, has
exactly the same number of points. Furthermore, every sphere, regardless
of size, has the same number of points as the entire space.

We remove the north pole of any sphere with a radius
of one inch, Cartesian Space point: (0.0, 1.0, 0.0)
we can see that no possible rotation will ever fill
in this missing point.

Here's another area where you flaunt your ignorance. Some of the pieces
used in the Banach-Tarski must not have a well-defined volume. Your
pieces above do have well-defined volumes.

TimePoint: 16:09-16:57
The gain of points in the video is an artifact of
the naming conventions. When we rotate the sphere
some of the original points take on new names, and
some of the new names refer to points that do not
actually exist within this subset of points.

The points exist.

Martin Shobe

[peteolcott]

Post by peteolcott
All of this is merely an artifact of his convoluted
naming convention.

"I don't understand it" is not a proof that the conclusion is wrong.

http://youtu.be/s86-Z-CbaHA
TimePoint: 16:48-16:55
"We have turned less than a quarter of the
  sphere into nearly three quarters just by rotating it."
When we try the same thing using a much simpler and
unequivocal naming convention, we find that no sphere
can possibly gain any points simply by rotating it.

Like many things that you choose to pontificate about, you are woefully ignorant about points in spheres. Every sphere, regardless of size, has exactly the same number of points. Furthermore, every sphere, regardless of size, has the same number of points
as the entire space.

Every sphere has an uncountably infinite number of points.

Hilbert's Hotel can be applied to a circle:
TimePoint 8:09-9:18
No hole will be left unfilled.

The mistake here is that at every point in time there is an
unfilled whole, thus the object has permanently ceased to
be a sphere by the removal of a single point.

The same thing is occurring in Banach–Tarski.
As soon as points are rotated to fill in gaps they create new gaps.
Simply ignoring these gaps does not make them actually go away.

Copyright 2018 Pete Olcott

We remove the north pole of any sphere with a radius
of one inch, Cartesian Space point: (0.0, 1.0, 0.0)
we can see that no possible rotation will ever fill
in this missing point.

Here's another area where you flaunt your ignorance. Some of the pieces used in the Banach-Tarski must not have a well-defined volume. Your pieces above do have well-defined volumes.

TimePoint: 16:09-16:57
The gain of points in the video is an artifact of
the naming conventions. When we rotate the sphere
some of the original points take on new names, and
some of the new names refer to points that do not
actually exist within this subset of points.

The points exist.
Martin Shobe

[Peter Percival]

Post by peteolcott

Post by Jim Burns

Post by peteolcott

Post by peteolcott

What do you mean by "georeferenced"?  It is not a word
I have come across before.

The center of a one inch diameter sphere is exactly ten
miles above the center of the north pole.
Without this degree of specificity the sphere is never
uniquely identified thus indiscernible from other spheres.

Let the center of a one-inch-diameter sphere S be ten miles
above the north pole. _Rigidly translate_ the points in
sphere S to another location such that the new center of S
is ten miles above the _south_ pole.

It thus ceases to be the original sphere at all.

Apparently you got the idea from somewhere that
Banach-Tarski claim is that the two resulting spheres are
in the same location as the one original sphere.
Have you considered finding out what the Banach-Tarski
theorem is? (BIG spoiler: It's not what you think it is.)

Think of it as the same idea as the Cantor's cardinality proof.

Why would one think of it as something that it is not? No wonder you
have spent 30 years learning nothing. Can you multiply 2 by 3? Olcott:
'think of it as adding 2 and 3'.

Post by peteolcott
We have an original sphere that has its own set of unique
points. A set of points that was previously not mapped to
any other points after decomposition and recomposition becomes
mapped to a whole new set of points that did not previously
exist. This proves that we really only just copied the sphere
without realizing it.
Copyright 2018 Pete Olcott

[peteolcott]

Post by Peter Percival

Post by peteolcott

Post by Jim Burns

Post by peteolcott

Post by peteolcott

What do you mean by "georeferenced"?  It is not a word
I have come across before.

The center of a one inch diameter sphere is exactly ten
miles above the center of the north pole.
Without this degree of specificity the sphere is never
uniquely identified thus indiscernible from other spheres.

Let the center of a one-inch-diameter sphere S be ten miles
above the north pole. _Rigidly translate_ the points in
sphere S to another location such that the new center of S
is ten miles above the _south_ pole.

It thus ceases to be the original sphere at all.

Apparently you got the idea from somewhere that
Banach-Tarski claim is that the two resulting spheres are
in the same location as the one original sphere.
Have you considered finding out what the Banach-Tarski
theorem is? (BIG spoiler: It's not what you think it is.)

Think of it as the same idea as the Cantor's cardinality proof.

Why would one think of it as something that it is not?

Even a very stupid computer program can be a mere naysayer.
I proved my point in the part you ignored.

Post by Peter Percival
No wonder you have spent 30 years learning nothing.  Can you multiply 2 by 3?  Olcott: 'think of it as adding 2 and 3'.

Post by peteolcott
We have an original sphere that has its own set of unique
points. A set of points that was previously not mapped to
any other points after decomposition and recomposition becomes
mapped to a whole new set of points that did not previously
exist. This proves that we really only just copied the sphere
without realizing it.
Copyright 2018 Pete Olcott

[Peter Percival]

Post by peteolcott

Post by Peter Percival

Post by peteolcott

Post by Jim Burns

Post by peteolcott

Post by peteolcott

What do you mean by "georeferenced"?  It is not a word
I have come across before.

The center of a one inch diameter sphere is exactly ten
miles above the center of the north pole.
Without this degree of specificity the sphere is never
uniquely identified thus indiscernible from other spheres.

Let the center of a one-inch-diameter sphere S be ten miles
above the north pole. _Rigidly translate_ the points in
sphere S to another location such that the new center of S
is ten miles above the _south_ pole.

It thus ceases to be the original sphere at all.

Apparently you got the idea from somewhere that
Banach-Tarski claim is that the two resulting spheres are
in the same location as the one original sphere.
Have you considered finding out what the Banach-Tarski
theorem is? (BIG spoiler: It's not what you think it is.)

Think of it as the same idea as the Cantor's cardinality proof.

Why would one think of it as something that it is not?

Even a very stupid computer program can be a mere naysayer.
I proved my point in the part you ignored.

No, I read it. How about you reading a proof of the Banach–Tarski
paradox? Actually, it would be a major achievement if you just learned
what it said, never mind the proof.

Post by peteolcott

Post by Peter Percival
 No wonder you have spent 30 years learning nothing.  Can you multiply
2 by 3?  Olcott: 'think of it as adding 2 and 3'.

Post by peteolcott
We have an original sphere that has its own set of unique
points. A set of points that was previously not mapped to
any other points after decomposition and recomposition becomes
mapped to a whole new set of points that did not previously
exist. This proves that we really only just copied the sphere
without realizing it.
Copyright 2018 Pete Olcott

[Peter Percival]

Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is

What?

Also, are you posting to appropriate newsgroups?

Post by peteolcott
that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Copyright 2018 Pete Olcott

[Peter Percival]

Post by Peter Percival

Post by peteolcott
Banach-Tarski Paradox
First stated in 1924, the Banach-Tarski paradox states that it is
possible to decompose a ball into six pieces which can be reassembled
by rigid motions to form two balls of the same size as the original.
The number of pieces was subsequently reduced to five by Robinson
(1947), although the pieces are extremely complicated.
I contend that it is impossible to decompose any sphere into any
subsets and recompose two identical spheres from these same points.
My proof of that is

What?
Also, are you posting to appropriate newsgroups?

Meant inappropriate.

Post by Peter Percival

Post by peteolcott
that when these points are georeferenced
(to eliminate the possibility of inadvertently using the same
points twice) it is self-evident that no method for achieving
the desired end-result can possibly exist.
On the other hand, not having these points georeferenced would
allow the possibility of using the same points twice to go
undetected.
Copyright 2018 Pete Olcott

[peteolcott]

The only system that I found that would uniquely
identify a single sphere is georeferencing.

If you had finished watching the video I directed you to, you'd see that that is pretty much exactly what is done. Every point on the surface is given a unique location marker. Despite that, you can still separate the points in the sphere into two spheres with equal volume.
EFQ

Every single point of the original sphere has its own unique
georeferenced location in the universe thus giving this original
sphere a unique identity.

Sets of these points are moved around such that the original
georeferenced sphere magically gives birth to a whole other
sphere with its own distinct georeferenced identity.

We end up with two spheres each with their own georeferenced
identity by simply moving some of the points of the original
georeferenced sphere around.

As soon as we move any points of the original sphere it loses
its identity. If we keep track of all the changes to the
geolocation of all of these points**

We can trace through the exact error of Banach–Tarski.
We will catch either a duplication of the points, or a
change in size of the sphere.

** A one inch diameter sphere exactly ten miles straight
up from the north pole of the Earth.

Copyright 2016, 2017, and 2018 Pete Olcott

[Jim Burns]

Post by peteolcott

On Wednesday, August 22, 2018 at 10:29:47 AM UTC-4,

The only system that I found that would uniquely
identify a single sphere is georeferencing.

If you had finished watching the video I directed you to,
you'd see that that is pretty much exactly what is done.
Every point on the surface is given a unique location
marker. Despite that, you can still separate the points in
the sphere into two spheres with equal volume.

Every single point of the original sphere has its own unique
georeferenced location in the universe thus giving this
original sphere a unique identity.
Sets of these points are moved around such that the original
georeferenced sphere magically gives birth to a whole other
sphere with its own distinct georeferenced identity.
We end up with two spheres each with their own georeferenced
identity by simply moving some of the points of the original
georeferenced sphere around.
As soon as we move any points of the original sphere it loses
its identity. If we keep track of all the changes to the
geolocation of all of these points**
We can trace through the exact error of Banach–Tarski.
We will catch either a duplication of the points, or a
change in size of the sphere.

Boiled down, what you're saying is that, if you traced
through _how_ Banach-Tarski disassembles one sphere and
reassembles the pieces into two spheres, you would find
the error.

Do you see how this is not an argument, just a
re-iteration of your belief that there is an error?

I don't think you know what an argument is.
I suspect that you think that it is the use of (what
elsewhere is called) techno-babble that makes some
text an "argument".

[peteolcott]

Post by Jim Burns

Post by peteolcott

On Wednesday, August 22, 2018 at 10:29:47 AM UTC-4,

The only system that I found that would uniquely
identify a single sphere is georeferencing.

If you had finished watching the video I directed you to,
you'd see that that is pretty much exactly what is done.
Every point on the surface is given a unique location
marker. Despite that, you can still separate the points in
the sphere into two spheres with equal volume.

Every single point of the original sphere has its own unique
georeferenced location in the universe thus giving this
original sphere a unique identity.
Sets of these points are moved around such that the original
georeferenced sphere magically gives birth to a whole other
sphere with its own distinct georeferenced identity.
We end up with two spheres each with their own georeferenced
identity by simply moving some of the points of the original
georeferenced sphere around.
As soon as we move any points of the original sphere it loses
its identity. If we keep track of all the changes to the
geolocation of all of these points**
We can trace through the exact error of Banach–Tarski.
We will catch either a duplication of the points, or a
change in size of the sphere.

Boiled down, what you're saying is that, if you traced
through _how_ Banach-Tarski disassembles one sphere and
reassembles the pieces into two spheres, you would find
the error.
Do you see how this is not an argument, just a
re-iteration of your belief that there is an error?
I don't think you know what an argument is.
I suspect that you think that it is the use of (what
elsewhere is called) techno-babble that makes some
text an "argument".

I watched the video.
I simplified my position and possibly caught the actual
Banach-Tarski error. To (identity of indiscernibles) identify
the sphere, we place a one inch sphere such that its center
is at the same point as the center of the Earth.

Now we have a unique sphere in the universe and can identify
each of its points in x,y,z Cartesian space.

If we remove as much as one circle from this sphere it
ceases to be spherical because of the infinitesimal gap
left in the original sphere.

The error of Banach-Tarski might seem to be the conflation
of no measurable gap (infinitesimal gap) with no gap at all.

Copyright 2018 Pete Olcott

[Peter Percival]

Post by peteolcott
The error of Banach-Tarski might seem to be the conflation
of no measurable gap (infinitesimal gap) with no gap at all.

The non-measurable sets that the proof of the Banach–Tarski paradox
speaks of are non-measurable in a technical sense that has nothing to do
with infinitesimals.

You know nothing of measure theory or infinitesimals and you don't have
the intellect to learn about either.

Post by peteolcott
Copyright 2018 Pete Olcott

[peteolcott]

Post by peteolcott
The error of Banach-Tarski might seem to be the conflation
of no measurable gap (infinitesimal gap) with no gap at all.

The non-measurable sets that the proof of the Banach–Tarski paradox speaks of are non-measurable in a technical sense that has nothing to do with infinitesimals.
You know nothing of measure theory or infinitesimals and you don't have the intellect to learn about either.

Post by peteolcott
Copyright 2018 Pete Olcott

Removing a single integer from the set of integers and it ceases to be the set of integers.

Removing a single circle from a sphere and it ceases to be a sphere.

Removing a single point from a sphere and it ceases to be a sphere,
it now has an infinitesimal gap and is thereby no longer perfectly spherical.

Copyright 2018 Pete Olcott.

[peteolcott]

Post by peteolcott

Post by Peter Percival

Post by peteolcott
The error of Banach-Tarski might seem to be the conflation
of no measurable gap (infinitesimal gap) with no gap at all.

The non-measurable sets that the proof of the Banach–Tarski
paradox speaks of are non-measurable in a technical sense
that has nothing to do with infinitesimals.
You know nothing of measure theory or infinitesimals and
you don't have the intellect to learn about either.

Removing a single integer from the set of integers and
it ceases to be the set of integers.
Removing a single circle from a sphere and it ceases to be
a sphere.
Removing a single point from a sphere and it ceases to be
a sphere, it now has an infinitesimal gap and is thereby
no longer perfectly spherical.

<PO>
 > All paradox necessarily has hidden error at its core.
 > You may not have the capacity to understand this.
</PO>
You (PO) seem to be applying your KEY hypothesis.
_Finding_ an error is not a matter of being first to
_claim_ to have found an error.

Finding an error can be accomplished by entirely ignoring
all of the reasoning and merely shown that the conclusion
is impossibly correct.

That seems to be what you're doing. You mention some
things that would be errors if they were done (losing
track of some of the points), and then you act as though
your job is done.
Does Banach-Tarski do any of these things? You apparently
don't see this as a question worth asking or answering.

As soon as a single point is removed from a sphere the
object utterly ceases to be a sphere. There are no rotations
to put that point back into the sphere.

That Banach-Tarski claims otherwise proves that Banach-Tarski
is wrong.

I could make similar remarks about your approach to
Godel's incompleteness or Tarski's undefinability of truth.
You make some claim and then act as though your job is done.

As soon as the conclusion of the 1931 GIT is proven to be
impossibly correct the whole thing has been utterly refuted
with out any need to look at any detail regarding how this
conclusion was formed.

This is the origin of my theory that _you don't know_
_what your job is_ , that you (PO) don't know what an
argument is, nor how to make a good one (or even a poor one).

Copyright 2018 Pete Olcott

[Peter Percival]

Post by peteolcott
Finding an error can be accomplished by entirely ignoring
all of the reasoning and merely shown that the conclusion
is impossibly correct.

You haven't shown that the conclusion is wrong.

[peteolcott]

Post by Peter Percival

Post by peteolcott
Finding an error can be accomplished by entirely ignoring
all of the reasoning and merely shown that the conclusion
is impossibly correct.

You haven't shown that the conclusion is wrong.

When we anchor the sphere in georeferencing such that its center
is at the center of the Earth, and this sphere has a one inch radius,
with the poles of the sphere determined by a line intersecting the
center of the poles of the Earth, then we can refer to the North pole
of the sphere as (0,1,0).

http://youtu.be/s86-Z-CbaHA

There is no possible rotation of this sphere that will fill in
the infinitesimal gap, thus the object ceases to be a sphere.
The video explained how these infinitesimal gaps were filled
in simply by rotating the sphere. With the simpler point naming
scheme this is shown to be obviously impossible.

Copyright 2018 Pete Olcott

[peteolcott]

Post by Peter Percival

Post by peteolcott
Finding an error can be accomplished by entirely ignoring
all of the reasoning and merely shown that the conclusion
is impossibly correct.

You haven't shown that the conclusion is wrong.

Just did it better now. This latest version is
clearer.

[Peter Percival]

Post by peteolcott
We can trace through the exact error of Banach–Tarski.
We will catch either a duplication of the points, or a
change in size of the sphere.

I don't know who this "we" is, but if you can get on with it.
Banach-Tarski's theorem has proofs in English. Find one and identify
where the first error is.

[peteolcott]

Post by peteolcott

The only system that I found that would uniquely
identify a single sphere is georeferencing.

If you had finished watching the video I directed you to, you'd see that that is pretty much exactly what is done. Every point on the surface is given a unique location marker. Despite that, you can still separate the points in the sphere into two spheres with equal volume.

Every single point of the original sphere has its own unique
georeferenced location in the universe thus giving this original
sphere a unique identity.

Which doesn't change the fact that the two ending spheres have the same volume as the original sphere. You keep using "identical" to mean something not intended by Banach-Tarski.
EFQ

The proof on the Banach-Tarski wiki is somewhat unsatisfying.
With Cayley, all the left rotations are the same group members
as all the right rotations.
Then, saying "there are unountably many starting points to the
countable members of the rotations", is unsatisfactory because
that doesn't guarantee uniqueness, either.
So, it is coming across wrong.
http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf
With for example "Grace a une remarque due a M.Lindenbaum,
on peut enoncer un theoreme analogue au lemme precedent
pour les ensembles lineaires, in remplacant le term "segment"
par "point"."
Vitali "we have non-measurable sets"
Lindenbaum "hi these are Veronese's"
Hausdorff "I spun Vitali's around for a 2-D analogue"
that Banach Tarski are coming across then
as "there's certainly enough here to find
some non-measurable ones then what
Hausdorff said holds and it doesn't matter
that our actual constructions don't add up
because they're non-measurable because
they're non-measurable there are uncountably
many to get them all and Vitali points out how
they double up".
So, the development on the Banach-Tarski wiki
is not Banach and Tarski's.
"Comme l' indique M.Hausdorff (en utilisant
une idee M.Vitali) on peut decompose tout
segment en une infinite nombrable de sous-
ensembles disjoints equivalents par decomposition
finite deux a deux. Soient: ..."
-- http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf

Removing a single point from a Sphere and it utterly ceases to be a sphere.

[Jeff Barnett]

Post by peteolcott

On Thursday, August 23, 2018 at 4:35:05 PM UTC-7,

Post by peteolcott

The only system that I found that would uniquely
identify a single sphere is georeferencing.

If you had finished watching the video I directed you to, you'd see
that that is pretty much exactly what is done. Every point on the
surface is given a unique location marker. Despite that, you can
still separate the points in the sphere into two spheres with equal
volume.

Every single point of the original sphere has its own unique
georeferenced location in the universe thus giving this original
sphere a unique identity.

Which doesn't change the fact that the two ending spheres have the
same volume as the original sphere. You keep using "identical" to
mean something not intended by Banach-Tarski.
EFQ

The proof on the Banach-Tarski wiki is somewhat unsatisfying.
With Cayley, all the left rotations are the same group members
as all the right rotations.
Then, saying "there are unountably many starting points to the
countable members of the rotations", is unsatisfactory because
that doesn't guarantee uniqueness, either.
So, it is coming across wrong.
http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf
With for example "Grace a une remarque due a M.Lindenbaum,
on peut enoncer un theoreme analogue au lemme precedent
pour les ensembles lineaires, in remplacant le term "segment"
par "point"."
   Vitali "we have non-measurable sets"
   Lindenbaum "hi these are Veronese's"
   Hausdorff "I spun Vitali's around for a 2-D analogue"
that Banach Tarski are coming across then
as "there's certainly enough here to find
some non-measurable ones then what
Hausdorff said holds and it doesn't matter
that our actual constructions don't add up
because they're non-measurable because
they're non-measurable there are uncountably
many to get them all and Vitali points out how
they double up".
So, the development on the Banach-Tarski wiki
is not Banach and Tarski's.
"Comme l' indique M.Hausdorff (en utilisant
une idee M.Vitali) on peut decompose tout
segment en une infinite nombrable de sous-
ensembles disjoints equivalents par decomposition
finite deux a deux.  Soient: ..."
  -- http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf

Removing a single point from a Sphere and it utterly ceases to be a sphere.

Watch! At no time do by fingers leave my hand: Remove a point from a
sphere (for whatever definition of point and sphere) then move the
remainder of the sphere to a new place, reinsert the point, and you
again have a sphere. If you agree to this you now have all of the basic
common sense understanding you need to follow Tarski. But you need a
bunch of math to really understand. You will, of course, remain ignorant
of the hard stuff.

[peteolcott]

Post by peteolcott

Post by peteolcott

The only system that I found that would uniquely
identify a single sphere is georeferencing.

If you had finished watching the video I directed you to, you'd see that that is pretty much exactly what is done. Every point on the surface is given a unique location marker. Despite that, you can still separate the points in the sphere into two
spheres with equal volume.

Every single point of the original sphere has its own unique
georeferenced location in the universe thus giving this original
sphere a unique identity.

Which doesn't change the fact that the two ending spheres have the same volume as the original sphere. You keep using "identical" to mean something not intended by Banach-Tarski.
EFQ

The proof on the Banach-Tarski wiki is somewhat unsatisfying.
With Cayley, all the left rotations are the same group members
as all the right rotations.
Then, saying "there are unountably many starting points to the
countable members of the rotations", is unsatisfactory because
that doesn't guarantee uniqueness, either.
So, it is coming across wrong.
http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf
With for example "Grace a une remarque due a M.Lindenbaum,
on peut enoncer un theoreme analogue au lemme precedent
pour les ensembles lineaires, in remplacant le term "segment"
par "point"."
   Vitali "we have non-measurable sets"
   Lindenbaum "hi these are Veronese's"
   Hausdorff "I spun Vitali's around for a 2-D analogue"
that Banach Tarski are coming across then
as "there's certainly enough here to find
some non-measurable ones then what
Hausdorff said holds and it doesn't matter
that our actual constructions don't add up
because they're non-measurable because
they're non-measurable there are uncountably
many to get them all and Vitali points out how
they double up".
So, the development on the Banach-Tarski wiki
is not Banach and Tarski's.
"Comme l' indique M.Hausdorff (en utilisant
une idee M.Vitali) on peut decompose tout
segment en une infinite nombrable de sous-
ensembles disjoints equivalents par decomposition
finite deux a deux.  Soient: ..."
  -- http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf

Removing a single point from a Sphere and it utterly ceases to be a sphere.

Watch! At no time do by fingers leave my hand: Remove a point from a sphere (for whatever definition of point and sphere) then move the remainder of the sphere to a new place, reinsert the point, and you again have a sphere. If you agree to this you now
have all of the basic common sense understanding you need to follow Tarski. But you need a bunch of math to really understand. You will, of course, remain ignorant of the hard stuff.

That is NOT what is happening.
http://youtu.be/s86-Z-CbaHA

At 15:00 he defines and removes six subsets of points.
and then later on takes 1/6 of the points of the sphere
and is able to add many points to this sphere by merely
rotating it and nothing else so that it now has 75% of
the original points.

All of this is merely an artifact of his convoluted
naming convention. He simply confuses himself into
believing that actual points pop into existence by
magic, simply by rotating the sphere.

If we derive another sphere with a simpler naming convention.
The point named by the Cartesian space xyz = (0,1,0) is
removed leaving a subset of the original sphere. There is
no possible rotation that ever puts this missing point back.

This unequivocally proves that the video version of
Banach–Tarski paradox is erroneous.

Copyright Pete Olcott 2018.

[peteolcott]

Post by peteolcott

The only system that I found that would uniquely
identify a single sphere is georeferencing.

If you had finished watching the video I directed you to, you'd see that that is pretty much exactly what is done. Every point on the surface is given a unique location marker. Despite that, you can still separate the points in the sphere into two spheres with equal volume.

Every single point of the original sphere has its own unique
georeferenced location in the universe thus giving this original
sphere a unique identity.

Which doesn't change the fact that the two ending spheres have the same volume as the original sphere. You keep using "identical" to mean something not intended by Banach-Tarski.
EFQ

Here is a much better argument:
http://youtu.be/s86-Z-CbaHA

At 15:00 he defines and removes six subsets of points.
and then later on takes 1/6 of the points of the sphere
and is able to add many points to this sphere by merely
rotating it and nothing else so that it now has 75% of
the original points.

All of this is merely an artifact of his convoluted
naming convention. He simply confuses himself into
believing that actual points pop into existence by
magic, simply by rotating the sphere.

If we derive another sphere with a simpler naming convention.**
The point named by the Cartesian space xyz = (0,1,0) is
removed leaving a subset of the original sphere. There is
no possible rotation that ever puts this missing point back.

If we remove a single point from any sphere the infinitesimal
gap causes it to utterly cease to be an actual sphere.

This unequivocally proves that the video version of
Banach–Tarski paradox is erroneous.

** A sphere with a one inch radius is geolocated such that
the center of this sphere is at the center of the Earth.
The poles of this sphere are defined by the line intersecting
the center of the poles of The Earth. {x,y,z} point {0,1,0}
is the north pole of this sphere.

Copyright Pete Olcott 2018.