Arbitrariness in set theory

[c***@gmail.com]

*Every* entry of the Cantor-list differs from the antidiagonal. From this it is concluded that *all* entries differ from the antidiagonal and therefore the antidiagonal is not in the list.

*Every* digit of an entry or of the antidiagonal is insufficient to determine a real number. From this it is not concluded that *all* digits of an entry or of the antidiagonal are insufficient to determine a real number.

*Every* rational number can be indexed by a natural number. From this it is concluded that *all* rational numbers can be indexed by natural numbers.
*Every* natural number leaves the overwhelming majority of rationals without index. From this it is not concluded that *all* natural numbers leave the overwhelming majority of rationals without index.

[William Elliot]

Post by c***@gmail.com
*Every* entry of the Cantor-list differs from the antidiagonal. From
this it is concluded that *all* entries differ from the antidiagonal
and therefore the antidiagonal is not in the list.

Ok.

Post by c***@gmail.com
*Every* digit of an entry or of the antidiagonal is insufficient to
determine a real number. From this it is not concluded that *all*
digits of an entry or of the antidiagonal are insufficient to
determine a real number.

That's not the point of Cantor's arguement.

Post by c***@gmail.com
*Every* rational number can be indexed by a natural number. From
this it is concluded that *all* rational numbers can be indexed by
natural numbers.

Ok.

Post by c***@gmail.com
*Every* natural number leaves the overwhelming
majority of rationals without index.

That depends upon the indexing.

Post by c***@gmail.com
From this it is not concluded that *all* natural numbers
leave the overwhelming majority of rationals without index.

I suppose.

[c***@gmail.com]

Post by William Elliot

Post by c***@gmail.com
*Every* entry of the Cantor-list differs from the antidiagonal. From
this it is concluded that *all* entries differ from the antidiagonal
and therefore the antidiagonal is not in the list.

Ok.

Post by c***@gmail.com
*Every* digit of an entry or of the antidiagonal is insufficient to
determine a real number. From this it is not concluded that *all*
digits of an entry or of the antidiagonal are insufficient to
determine a real number.

That's not the point of Cantor's arguement.

Cantor claims that real numbers are defined by their digits. That is impossible without finite formula or end signal

Post by William Elliot

Post by c***@gmail.com
*Every* rational number can be indexed by a natural number. From
this it is concluded that *all* rational numbers can be indexed by
natural numbers.

Ok.

Post by c***@gmail.com
*Every* natural number leaves the overwhelming
majority of rationals without index.

That depends upon the indexing.

No, that is true for every indexing. If you disagree try to find an index that does not leavesthe overwhelming majority of rationals without index.

[William Elliot]

Post by c***@gmail.com

Post by William Elliot

Post by c***@gmail.com
*Every* entry of the Cantor-list differs from the antidiagonal.
From this it is concluded that *all* entries differ from the
antidiagonal and therefore the antidiagonal is not in the list.

Ok.

Cantor claims that real numbers are defined by their digits. That is
impossible without finite formula or end signal

A sequence of binary digits is a function from N into { 0,1 }.
It is infinite. What you're showing is that finite minded
computer limited microbrains can't handle Cantor.

Post by c***@gmail.com

Post by William Elliot

Post by c***@gmail.com
*Every* rational number can be indexed by a natural number. From
this it is concluded that *all* rational numbers can be indexed by
natural numbers.

Ok.

Post by c***@gmail.com
*Every* natural number leaves the overwhelming
majority of rationals without index.

That depends upon the indexing.

No, that is true for every indexing. If you disagree try to find an index that does not leavesthe overwhelming majority of rationals without index.

It is well known that N^2 is equinumerous to N.
Also Q is equinumerous to N^2. Thus Q is equinumerous to N.

[c***@gmail.com]

Post by William Elliot
It is well known that N^2 is equinumerous to N.
Also Q is equinumerous to N^2. Thus Q is equinumerous to N.

The contradiction, i.e., the proof of not enumerating the rational numbers can be found here on p. 258: https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

[William Elliot]

Post by c***@gmail.com

Post by William Elliot
It is well known that N^2 is equinumerous to N.
Also Q is equinumerous to N^2. Thus Q is equinumerous to N.

The contradiction, i.e., the proof of not enumerating the rational
https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

Oh him. Are you he?

[Me]

Post by c***@gmail.com
*Every* natural number leaves the overwhelming
majority of rationals without index.

Indeed! Now, for the sake of the argument, let's consider the set of natural numbers. We may "index" them by themselves. I.e. we just consider the identity function

f: IN --> IN with f(n) = n. :-)

With other words, we consider the mapping n <-> n for any n e IN.

If you disagree try to find an index that does not leave the overwhelming
majority of rationals without index.

Right. So your FINE ARGUMENT shows that the set of natural numbers isn't coutably infinite (enumerable) too! After all, "*every* natural number [used as an index] leaves the overwhelming majority of natural numbers without index." (I guess you are actually considering the indexed elements from index 1 up to some arbitrary index n, where n e IN).

A very clever argument, Wolfgang, indeed!

One might even try to rephrase it the following way:

"There are no infinite sets because *no* finite set is infinite."

WOW, how could we ever miss THAT?!

[c***@gmail.com]

Post by Me

Post by c***@gmail.com
*Every* natural number leaves the overwhelming
majority of rationals without index.

Indeed! Now, for the sake of the argument, let's consider the set of natural numbers. We may "index" them by themselves. I.e. we just consider the identity function
f: IN --> IN with f(n) = n. :-)
With other words, we consider the mapping n <-> n for any n e IN.

If you disagree try to find an index that does not leave the overwhelming
majority of rationals without index.

Right. So your FINE ARGUMENT shows that the set of natural numbers isn't coutably infinite (enumerable) too!

Of course not. The notion of countability is nonsense.

Post by Me
After all, "*every* natural number [used as an index] leaves the overwhelming majority of natural numbers without index." (I guess you are actually considering the indexed elements from index 1 up to some arbitrary index n, where n e IN).
A very clever argument, Wolfgang, indeed!

It is not refutable. The nonsense of countability is more drastically shown with the set of rational numbers, but it is as invalid for the natural numbers alone. There is no finised infinity. But without no completeness of countable sets can be shown.

Post by Me
"There are no infinite sets because *no* finite set is infinite."

And there are only finite sets, i.e., finite initial segments. No natural number is outside of all of them.

Post by Me
WOW, how could we ever miss THAT?!

Cantor has spread his insanity with great success.

[Me]

Post by c***@gmail.com
And there are only finite sets, i.e., finite initial segments.

This may be the case in the context of some "finite systems of mathematics", of course. See: https://en.wikipedia.org/wiki/Finitism

But in the context of classical mathematics / set theory it is simply WRONG (due to the AoI), see: https://en.wikipedia.org/wiki/Axiom_of_infinity.

See: https://en.wikipedia.org/wiki/Classical_mathematics

[c***@gmail.com]

Post by Me

Post by c***@gmail.com
And there are only finite sets, i.e., finite initial segments.

This may be the case in the context of some "finite systems of mathematics", of course. See: https://en.wikipedia.org/wiki/Finitism

No, it is also true in set theory: All rows of

1
1, 2
1, 2, 3
...

are finite. And moving a finite row into another position dos not change it. You are forced to deny translation invariance. For every observer not yet perverted by set theory this shows the perversion of set theory.

[Me]

Post by Me

Post by c***@gmail.com
And there are only finite sets, i.e., finite initial segments.

This may be the case in the context of some "finite systems of
mathematics", of course. See: https://en.wikipedia.org/wiki/Finitism

No, it is also true in set theory [...].

No, it's NOT true in set theory, say ZFC, due to the AoI:

"In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers."

Source: https://en.wikipedia.org/wiki/Axiom_of_infinity

Post by Me
in the context of classical mathematics/set theory [your claim] is simply
WRONG (due to the AoI), see https://en.wikipedia.org/wiki/Axiom_of_infinity.

[c***@gmail.com]

Post by Me

Post by c***@gmail.com
And there are only finite sets, i.e., finite initial segments.

This may be the case in the context of some "finite systems of
mathematics", of course. See: https://en.wikipedia.org/wiki/Finitism

No, it is also true in set theory [...].

Either you have not understood what the axiom saysw or you are lying. All rows of

1
1, 2
1, 2, 3
...

are finite.

Further they remain finite when written in another place, for instance when shifted into the first row.

[Me]

Post by Me

[...] there are only finite sets, i.e., finite initial segments.

This may be the case in the context of some "finite systems of
mathematics", of course. See: https://en.wikipedia.org/wiki/Finitism

No, it is also true in set theory [...].

No, it's NOT true in set theory, say ZFC, due to the AoI.

Either <bla bla>

Hint:

"In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers."

https://en.wikipedia.org/wiki/Axiom_of_infinity

Just too dumb to understand basic set theory, Mucke?

[Me]

Post by Me

[...] there are only finite sets, i.e., finite initial segments.

This may be the case in the context of some "finite systems of
mathematics", of course. See: https://en.wikipedia.org/wiki/Finitism

No, it is also true in set theory [...].

No, it's NOT true in set theory, say ZFC, due to the AoI.

Either <bla bla>

Hint:

"[The axiom of infinity] guarantees the existence of at least one infinite set, namely a set containing the natural numbers."

https://en.wikipedia.org/wiki/Axiom_of_infinity

Just too dumb to understand basic set theory, Mucke?

[Julio Di Egidio]

<snip>

Post by Me

Either <bla bla>

"[The axiom of infinity] guarantees the existence of at least one infinite set, namely a set containing the natural numbers."
https://en.wikipedia.org/wiki/Axiom_of_infinity
Just too dumb to understand basic set theory, Mucke?

"Understand" as in read your authorities?

Neo-empire: an elite of insane criminals, then abusive retards all the way
down...

Julio

[Me]

Post by Julio Di Egidio

Post by Me
Just too dumb to understand basic set theory, Mucke?

"Understand" as in read your authorities?

No, but as in RTFM (i.e read some textbooks, man).

[Simon Roberts]

Post by Julio Di Egidio
<snip>

Post by Me

Either <bla bla>

"[The axiom of infinity] guarantees the existence of at least one infinite set, namely a set containing the natural numbers."
https://en.wikipedia.org/wiki/Axiom_of_infinity
Just too dumb to understand basic set theory, Mucke?

"Understand" as in read your authorities?
Neo-empire: an elite of insane criminals, then abusive retards all the way
down...

"abusive retards"

Post by Julio Di Egidio
Julio

[c***@gmail.com]

Post by Me
"[The axiom of infinity] guarantees the existence of at least one infinite set, namely a set containing the natural numbers."

The axiom of infinity does not guarantee that a finite set (FISON = finite initial segment of |N), i.e., a row of the list
1
1, 2
1, 2, 3
...
is infinite.

[Me]

Post by c***@gmail.com
The axiom of infinity does not guarantee that a finite set (FISON = finite
initial segment of IN), i.e., a [set] of the [following sequence of sets]

({1}, {1, 2}, {1, 2, 3}, ...)

Post by c***@gmail.com
is infinite.

Indeed. But in set theory, the union of (the set of) all FISONs, i. e. U {{1}, {1, 2}, {1, 2, 3}, ...}, is the INFINITE set {1, 2, 3, ...} = IN.

Hint: Since each and every natural number is in some or other FISON, each and every natural number also must be in the UNION of (the set of) all FISONs.

[c***@gmail.com]

Post by Me

Post by c***@gmail.com
The axiom of infinity does not guarantee that a finite set (FISON = finite
initial segment of IN), i.e., a [set] of the [following sequence of sets]

({1}, {1, 2}, {1, 2, 3}, ...)

Post by c***@gmail.com
is infinite.

Indeed.

Why then prattle that nonsense.

Post by Me
But in set theory, the union of (the set of) all FISONs, i. e. U {{1}, {1, 2}, {1, 2, 3}, ...}, is the INFINITE set {1, 2, 3, ...} = IN.
Hint: Since each and every natural number is in some or other FISON, each and every natural number also must be in the UNION of (the set of) all FISONs.

As we just have seen from translation invariance, the INTERPRETATION of infinite sets as something being finished and having a cardinal number is utter rubbish.

[Me]

As we just have seen from translation invariance, <bla>

I guess *you* are the only one being able to "see" anything "from translation invariance" (whatsoever). Hint: I's called "delusion".

[Me]

As we just have seen from translation invariance, <bla>

I guess *you* are the only one being able to "see" anything "from translation invariance" (whatsoever). Hint: It's called "delusion".

[c***@gmail.com]

Post by Me

Post by c***@gmail.com
As we just have seen from translation invariance

I guess *you* are the only one being able to "see" anything "from translation invariance"

Have you ever written an X when you voted by ballot and have you trusted that it would remain a single X? When polling you cannot be as sure as when translating the X from one to another place in mathematics.

[FredJeffries]

Post by c***@gmail.com

Post by Me

Post by c***@gmail.com
As we just have seen from translation invariance

I guess *you* are the only one being able to "see" anything "from translation invariance"

Have you ever written an X when you voted by ballot and have you trusted that it would remain a single X? When polling you cannot be as sure as when translating the X from one to another place in mathematics.

By "translation invariance", then, putting your "X" in the box labeled "Yes" is the same as putting it in the box labeled "No".

[Me]

Post by FredJeffries

Post by c***@gmail.com

Post by Me

Post by c***@gmail.com
As we just have seen from translation invariance

I guess *you* are the only one being able to "see" anything "from
translation invariance"

Have you ever written an X when you voted by ballot and have you trusted
that it would remain a single X? When polling you cannot be as sure as when
translating the X from one to another place in mathematics.

By "translation invariance", then, putting your "X" in the box labeled "Yes"
is the same as putting it in the box labeled "No".

It seems to me that it might have never occurred to him that the meaning of many (if not most) expressions is "context dependent" ...

For example, let's consider the expression

a = 1

and another expression, say,

1 = b .

If we shift up the latter, we might just end up with the statement

a = 11 = b .

Hmmm... "Translation invariance", huh?!

Another example. Let's consider the two statements:

It is true that

c = 0 . (*)

It is false that

d = 0 . (**)

Now let's just "move" (**) up two lines and (*) down two lines. This way we get:

It is true that

d = 0 . (**)

It is false that

c = 0 . (**)

"Translation invariance", sure ...

And how about the following?

{ }

obviously denotes the empty set.

And

1

the number one. By "Mückenheim's translation invariance"


{ 1 }

still denotes the empty set! :-)

Or how about this one: The following depicts (due to Mückeneheim) a sequence of 3 numbers (namely the numbers 1, 2 and 3) in the unary numeral system.

o
oo
ooo

Due to Mückenheim's principle of "translation invariance" this sequence is the same as the sequence

ooo

just consisting of the number 3 (or a unary representation of it).

Well...

"Translation invariance" might hold for a mathematical textbook (as a whole), but that hardly accounts for Mückenheim's "principle of translation invariance", I'd say.

[c***@gmail.com]

Post by Me
Or how about this one: The following depicts (due to Mückeneheim) a sequence of 3 numbers (namely the numbers 1, 2 and 3) in the unary numeral system.
o
oo
ooo
Due to Mückenheim's principle of "translation invariance" this sequence is the same as the sequence
ooo

No.

Post by Me
Well...
"Translation invariance" might hold for a mathematical textbook (as a whole), but that hardly accounts for Mückenheim's "principle of translation invariance", I'd say.

The question is whether finite rows when translated into the first row can yield an infinite row. The answer is no.

[Me]

Post by c***@gmail.com
The question is whether finite rows when translated into the first row
can yield an infinite row. [...]

If there were infinitely many rows (growing beyond all bounds) then one might argue that they can.

In (the context of) _set theory_ we might just consider the UNION of (the set of) the infinitely many FISONS {1}, {1, 2}, {1, 2, 3}, ...:

X = U {{1}, {1, 2}, {1, 2, 3}, ...}

or (more formally)

X = U {{1,...,n} : n e IN} .

It's rather simple to PROVE that X = IN. Hence X is infinite, since IN is infinite.

[c***@gmail.com]

Post by Me

Post by c***@gmail.com
The question is whether finite rows when translated into the first row
can yield an infinite row. [...]

If there were infinitely many rows (growing beyond all bounds) then one might argue that they can.

Of course, the result can grow beyond every bound - but it caqn never become a fixed quantity larger than all finite rows. Cantor and his disciples have confused these two different meanings of infinity. Something growing and something fixed.

Post by Me
X = U {{1}, {1, 2}, {1, 2, 3}, ...}
or (more formally)
X = U {{1,...,n} : n e IN} .
It's rather simple to PROVE that X = IN. Hence X is infinite, since IN is infinite.

Again this deliberate neglect of precision! The correct statement is: It's rather simple to PROVE that X = IN. Hence X is potentially infinite, since IN is potentially infinite.

[Me]

Post by c***@gmail.com

Post by Me
In (the context of) _set theory_ we might just consider the UNION of
X = U {{1}, {1, 2}, {1, 2, 3}, ...}
or (more formally)
X = U {{1,...,n} : n e IN} .
It's rather simple to PROVE that X = IN. Hence X is infinite, since IN is infinite.

Of course, I'm STILL arguing in the context of _set theory_ here, idiot. Note that I said _at the beginning_ of the argument:

In (the context of) _set theory_ ...

Post by c***@gmail.com
Again this deliberate neglect of precision!

*sigh*

[Me]

Post by Me
In (the context of) _set theory_ we might just consider the UNION of
X = U {{1}, {1, 2}, {1, 2, 3}, ...}
or (more formally)
X = U {{1,...,n} : n e IN} .
It's rather simple to PROVE that X = IN. Hence X is infinite, since IN

Here's a short proof for the latter claim:

Just consider the function f: IN --> {2, 3, 4, ...} with f(n) = n+1. f is a bijective function from IN onto {2, 3, 4, ...}. This proves that IN = {1, 2, 3, ...} is (Dedekind-)infinite (since {2, 3, 4, ...} is a proper subset of IN = {1, 2, 3, ...}).

Note: "In mathematics, a set A is (Dedekind-)infinite [...] if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A." (Wikipedia)

[Jack Campin]

***@gmail.com wrote: [shit]

Muck off, Fuckenheim.

-----------------------------------------------------------------------------
e m a i l : j a c k @ c a m p i n . m e . u k
Jack Campin, 11 Third Street, Newtongrange, Midlothian EH22 4PU, Scotland
mobile [phone] <http://www.campin.me.uk> Twitter: JackCampin

[c***@gmail.com]

Post by Jack Campin
Muck off, Fuckenheim.

Do you believe to impress me? You prove that some Cantor believers are - well, just like you.

[Simon Roberts]

Post by Me

Post by c***@gmail.com
*Every* natural number leaves the overwhelming
majority of rationals without index.

Indeed! Now, for the sake of the argument, let's consider the set of natural numbers. We may "index" them by themselves. I.e. we just consider the identity function
f: IN --> IN with f(n) = n. :-)

fuck you and your notation. deal with what is given don't deny what he stated based on your "transformation" it into something DIFFERENT, for the SAKE of your ... bullshit. Can you do this? Can you reach his level and argue fairly?

I would never expect an appropriate answer.

Post by Me
With other words, we consider the mapping n <-> n for any n e IN.

If you disagree try to find an index that does not leave the overwhelming
majority of rationals without index.

Right. So your FINE ARGUMENT shows that the set of natural numbers isn't coutably infinite (enumerable) too! After all, "*every* natural number [used as an index] leaves the overwhelming majority of natural numbers without index." (I guess you are actually considering the indexed elements from index 1 up to some arbitrary index n, where n e IN).
A very clever argument, Wolfgang, indeed!
"There are no infinite sets because *no* finite set is infinite."
WOW, how could we ever miss THAT?!

[Simon Roberts]

Post by William Elliot

Post by c***@gmail.com
*Every* entry of the Cantor-list differs from the antidiagonal. From
this it is concluded that *all* entries differ from the antidiagonal
and therefore the antidiagonal is not in the list.

Ok.

Post by c***@gmail.com
*Every* digit of an entry or of the antidiagonal is insufficient to
determine a real number. From this it is not concluded that *all*
digits of an entry or of the antidiagonal are insufficient to
determine a real number.

That's not the point of Cantor's arguement.

Post by c***@gmail.com
*Every* rational number can be indexed by a natural number. From
this it is concluded that *all* rational numbers can be indexed by
natural numbers.

Ok.

Post by c***@gmail.com
*Every* natural number leaves the overwhelming
majority of rationals without index.

That depends upon the indexing.

Fuck, sorry Claus, just index the positive integers as either the subset or as a subset of either reals or rationals with the natural numbers. DONE.

These, left over, rationals or reals, as stated, DO NOT HAVE AN INDEX.

(oops, yelling).

Post by William Elliot

Post by c***@gmail.com
From this it is not concluded that *all* natural numbers
leave the overwhelming majority of rationals without index.

and reals, as well, without an index.

Post by William Elliot
I suppose.

I apologize, I do not see the difficulty. Cantor's argument is based on an arbitrary set of infinite binary sequences. It only proves that there is an infinite set that cannot have a one-to-one correspondence with natural numbers. So?

My question would be, can you have an infinite subset of naturals that has a one to one correspondence with naturals.

(0,2,4,6,8,...) and (0,1,2,3,4,...) are two sets which match this description.

(1,2,3,4, ....) and (0,1,2,3,4,...) are two sets I intuitively believe cannot have a on-to-one correspondence. I cannot prove this true one way or the other.

If |N = (0,1,2,3, ...) and any given S_2 not empty and a subset of naturals then

If |N - S_2 = S_1 and either S_1 or S_2 is a finite set then there is no one to one correspondence between any of the sets.

can be proved like Cantor's.

So many more potential cases of correspondence or non correspondence.

[Simon Roberts]

Post by William Elliot

Post by c***@gmail.com
*Every* entry of the Cantor-list differs from the antidiagonal. From
this it is concluded that *all* entries differ from the antidiagonal
and therefore the antidiagonal is not in the list.

Ok.

Post by c***@gmail.com
*Every* digit of an entry or of the antidiagonal is insufficient to
determine a real number. From this it is not concluded that *all*
digits of an entry or of the antidiagonal are insufficient to
determine a real number.

That's not the point of Cantor's arguement.

Post by c***@gmail.com
*Every* rational number can be indexed by a natural number. From
this it is concluded that *all* rational numbers can be indexed by
natural numbers.

Ok.

Post by c***@gmail.com
*Every* natural number leaves the overwhelming
majority of rationals without index.

That depends upon the indexing.

Fuck, sorry Claus, just index the positive integers =I and I as either the subset or as a subset of either reals or rationals with the natural numbers. DONE.
These, left over, rationals or reals, as stated, DO NOT HAVE AN INDEX.
(oops, yelling).

Post by William Elliot

Post by c***@gmail.com
From this it is not concluded that *all* natural numbers
leave the overwhelming majority of rationals without index.

and reals, as well, without an index.

Post by William Elliot
I suppose.

I apologize, I do not see the difficulty. Cantor's argument is based on an "arbitrary" set of infinite binary sequences. It only proves that there is an infinite set (of sequences) that cannot have a one-to-one correspondence with (be indexed by) natural numbers. So?
My question would be, can you have an infinite subset of naturals that has a one to one correspondence with naturals.
(0,2,4,6,8,...) and (0,1,2,3,4,...) are two sets which match this description.
(1,2,3,4, ....) and (0,1,2,3,4,...) are two sets I intuitively believe cannot have a on-to-one correspondence. I can prove this.
If |N = (0,1,2,3, ...) and any given S_2 not empty and a subset of naturals, |N, then
If |N - S_2 = S_1 and either S_1 or S_2 is a finite set then there is no one to one correspondence between any of these three sets.
can be proved like Cantor's diagonal argument.
So many more potential cases of correspondence or non correspondence can be proved.

[Me]

Post by Simon Roberts
My question would be, can you have an infinite subset of naturals that has
a one to one correspondence with naturals.

Yes. Actually, this follows from the very definition of (Dedekind-)"infinite":

"In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A." (Wikipedia)

Post by Simon Roberts
(0,2,4,6,8,...) and (0,1,2,3,4,...) are two sets ...

Hence (using standard notation) I'd rather write {0, 2, 4, 6, 8} and {0, 1, 2, 3, 4, ...} in this case.

Post by Simon Roberts
(1,2,3,4, ....) and (0,1,2,3,4,...) are two sets I intuitively believe cannot
have a on-to-one correspondence. I cannot prove this true one way or the
other.

Ah? Just consider the bijective function f: IN --> {1, 2, 3, ...} with f(n) = n+1. f is a 1:1 correspondence between {0, 1, 2, 3, ...} and {1, 2, 3, 4, ...}. This proves that IN = {0, 1, 2, 3, ...} is a (Dedekind-)infinite set.

Again, we can now prove that {1, 2, 3, 4, ...} is (Dedekind-)infinite by considering the bijective function g: {1, 2, 3, ...} --> {2, 3, 4, ...} with g(n) = n+1. g is a 1:1 correspondence between {1, 2, 3, ...} and {2, 3, 4, ...}. Now this proves that {1, 2, 3, ...} (a proper subset of the natural numbers) is a (Dedekind-)infinite set.

Now we have established that there is an infinite subset of the set of natural numbers, namely {1, 2, 3, ...}, which "has a one to one correspondence with naturals" (namely f).

[c***@gmail.com]

Post by Me
Again, we can now prove that {1, 2, 3, 4, ...} is (Dedekind-)infinite by considering the bijective function g: {1, 2, 3, ...} --> {2, 3, 4, ...} with g(n) = n+1. g is a 1:1 correspondence between {1, 2, 3, ...} and {2, 3, 4, ...}. Now this proves that {1, 2, 3, ...} (a proper subset of the natural numbers) is a (Dedekind-)infinite set.
Now we have established that there is an infinite subset of the set of natural numbers, namely {1, 2, 3, ...}, which "has a one to one correspondence with naturals" (namely f).

Assuming that the set |N is Cantor infinite, the limit of the sequence {1}, {2}, {3}, ... is empty. Otherwise we could not apply all natural numbers for counting purposes or else. So we are forced to believe that we can exhaust the set |N. But we must not believe that we can exhaust the sequence {1}, {1}, {1}, ... of aleph_0 ones - unless we imagine that each one is indexed according to its position in the sequence. Then we can exhaust it.

Can there be something on earth that is as stupid as this idea?

[Me]

Assuming that the set IN is Cantor infinite, <bla and blub>

"Cantor infinite"? Mind to define that term?

[c***@gmail.com]

Post by Me

Assuming that the set IN is Cantor infinite, <bla and blub>

"Cantor infinite"? Mind to define that term?

His masters voice:

In der ersteren Form, als Uneigentlich-Unendliches, stellt es sich als ein veränderliches Endliches dar; in der andern Form, wo ich es


Eigentlich-unendliches


nenne, tritt es als ein durchaus bestimmtes Unendliches auf. Die unendlichen realen ganzen Zahlen, welche ich im folgenden definieren will und zu denen ich schon vor einer längeren Reihe von Jahren geführt worden bin, ohne daß es mir zum deutlichen Bewußtsein gekommen war, in ihnen konkrete Zahlen von realer Bedeutung zu besitzen, haben durchaus nichts gemein mit der ersteren von beiden Formen, mit dem Uneigentlich-unendlichen, dagegen ist ihnen derselbe Charakter der Bestimmtheit eigen, wie wir ihn bei dem unendlich fernen Punkt in der analytischen Funktionentheorie antreffen; sie gehören also zu den Formen und Affektionen des Eigentlich-unendlichen.

Gesammelte Abh. S. 166

[Me]

Post by Me

Assuming that the set IN is Cantor infinite, <bla and blub>

"Cantor infinite"? Mind to define that term?

His masters voice: <bla>

So where's the definition I asked for?

You know, a statement of the form

A set S is /Cantor infinite/ if ...

Mind to deliver such a definition?

[c***@gmail.com]

Post by Me

Post by Me

Assuming that the set IN is Cantor infinite, <bla and blub>

"Cantor infinite"? Mind to define that term?

So where's the definition I asked for?
You know, a statement of the form
A set S is /Cantor infinite/ if ...
Mind to deliver such a definition?

The text deleted by you contained the phrase: "Die unendlichen realen ganzen Zahlen, welche ich im folgenden definieren will ..." This is taken from p. 166 of Cantors collected works. Therefore you should be able to find Cantor's definition.

However, "obviously it was difficult for Cantor to express in hard mathematical language what he imagined. His 'definition' could appear rather questionable to a critical thinker like Kronecker." [Herbert Meschkowski: "Georg Cantor: Leben, Werk und Wirkung", 2nd ed., Bibl. Inst., Mannheim (1983) pp. 213 & 229]

[Me]

Post by c***@gmail.com

Post by Me

Post by Me

Assuming that the set IN is Cantor infinite, <bla and blub>

"Cantor infinite"? Mind to define that term?

So where's the definition I asked for?
A set S is /Cantor infinite/ if ...
Mind to deliver such a definition?

The text deleted by you contained the phrase: "Die unendlichen realen ganzen
Zahlen, welche ich im folgenden definieren will ..."

I didn't ask for a discussion of Cantor's phrase "unendliche reale ganze Zahlen" or whatever. I asked for a definition of a /Cantor infinite/ set. Something that YOU mentioned, idiot.

Why the the hell are you ALWAYS using terms you cannot even define?!

[c***@gmail.com]

Post by Me

Post by c***@gmail.com

Post by Me

Post by Me

Assuming that the set IN is Cantor infinite, <bla and blub>

"Cantor infinite"? Mind to define that term?

So where's the definition I asked for?
A set S is /Cantor infinite/ if ...
Mind to deliver such a definition?

The text deleted by you contained the phrase: "Die unendlichen realen ganzen
Zahlen, welche ich im folgenden definieren will ..."

I didn't ask for a discussion of Cantor's phrase "unendliche reale ganze Zahlen" or whatever. I asked for a definition of a /Cantor infinite/ set. Something that YOU mentioned,

He defines it. Look here. Hint: It is a set that can be exhausted.

[Me]

Post by c***@gmail.com

Post by Me
I didn't ask for a discussion of Cantor's phrase "unendliche reale ganze
Zahlen" or whatever. I asked for a definition of a /Cantor infinite/ set.
Something that YOU mentioned,

He defines it.

Look idiot, I asked *you* for a definition of a "term" *you* used.

With other words, I'm asking for a statement of the form:

A set S is /Cantor infinite/ iff ...

Post by c***@gmail.com
Hint: It is a set that can be exhausted.

So your definition is

A set S is /Cantor infinite/ iff it can be exhausted. (*)

Right? Now, what's the definition of "exhausted"? Without defining that term your "definition" (*) is just moot.

Again, I'm asking for a statement of the form:

A set S /can be exhausted/ iff ...

Actually, I'd prefer a definition of a "property". We might call it "exhaustible". So please give me a definition:

A set S is /exhaustible/ iff ...

But then it seems to me that /exhaustible/ and /Cantor infinite/ just mean the same; after all in this case we would have:

For any S: S is /Cantor infinite/ iff it is /exhaustible/.

So please... DEFINE *either* /Cantor infinite/ OR /exhaustible/:

a)
A set S is /Cantor infinite/ iff ...

b)
A set S is /exhaustible/ iff ...

Of course, you should use (proper) SET THEORETIC terms for your definition, after all the context of the present discussion is SET THEORY.

=========================

An afterthought. Maybe you just meant /infinite/ when you wrote

"Assuming that the set IN is Cantor infinite ..."

Well, why didn't you say so in the first place?

Note though that in the context of SET THEORY we do not have to "assume" that, since we can PROVE that IN (defined as usual in this context) is /infinite/. In addition there is no need to define /infinite/ in this context, since the term is "well known" and properly defined in the context of, say, ZFC.

See: http://mathworld.wolfram.com/InfiniteSet.html

[Me]

Post by Me
But then it seems to me that /exhaustible/ and /Cantor infinite/ just mean
For any S: S is /Cantor infinite/ iff it is /exhaustible/.
a)
A set S is /Cantor infinite/ iff ...
b)
A set S is /exhaustible/ iff ...
Of course, you should use (proper) SET THEORETIC terms for your definition,
after all the context of the present discussion is SET THEORY.

Another afterthought. Maybe you just meant /countable infinite/ when you wrote

"Assuming that the set [...] is Cantor infinite ..."

At least this would have a certain relation to the set of natural numbers and "exhaustibility": :-) After all:

"An infinite set whose elements can be put into a one-to-one correspondence with the set of [natural numbers] is said to be /countably infinite/."

Source: http://mathworld.wolfram.com/InfiniteSet.html (slightly modified)

[WM]

Post by Me

Post by c***@gmail.com

Post by Me
I didn't ask for a discussion of Cantor's phrase "unendliche reale ganze
Zahlen" or whatever. I asked for a definition of a /Cantor infinite/ set.
Something that YOU mentioned,

He defines it.

I asked *you* for a definition of a "term" *you* used.

And I told you where you can find the definition.

Post by Me
A set S is /Cantor infinite/ iff ...

if it is defined as Cantor did.

Post by Me

Post by c***@gmail.com
Hint: It is a set that can be exhausted.

So your definition is
A set S is /Cantor infinite/ iff it can be exhausted. (*)

An infinite set is Cantor infinite if it can be exhausted.

Post by Me
Right? Now, what's the definition of "exhausted"? Without defining that term your "definition" (*) is just moot.

There is always a word remaining that you cannot understand. The reason is that every definition requires words whcih have no meaning for illiterate persons.

Post by Me
A set S /can be exhausted/ iff ...
A set S is /exhaustible/ iff ...
But then it seems to me that /exhaustible/ and /Cantor infinite/ just mean the same;

How did you come to this hidden conclusion? Have had some help?

Post by Me
For any S: S is /Cantor infinite/ iff it is /exhaustible/.
a)
A set S is /Cantor infinite/ iff ...
b)
A set S is /exhaustible/ iff ...

A set is exhaustible if it is X. But of course you will ask what X is. Therefore I do not insert it.

Post by Me
Of course, you should use (proper) SET THEORETIC terms for your definition, after all the context of the present discussion is SET THEORY.

Set theorists call it infinite and refuse to understand the difference between really infinite and Cantor infinite because then they would get trapped in the mud of their nonsense.

Post by Me
=========================
"Assuming that the set IN is Cantor infinite ..."
Well, why didn't you say so in the first place?
Note though that in the context of SET THEORY we do not have to "assume" that, since we can PROVE that IN (defined as usual in this context) is /infinite/.

That's the point. WE can prove that it is potentially infinite. Nobod can prove that it is Cantor-infinite. But you refuse to understand the correct meaning of infinite and decide to believe nonsense like finished infinity. Transfinite set theory is simply a matter of deliberately blinding oneself.

Regards, WM

[Me]

Nobod[y] can prove that [IN] is [Dedekind-]infinite.

Wrong, Mückeheim. Actually it's rather easy to prove that (in the context of set theory, that is).

Look, here's a rather short proof:

Just consider the function f: IN --> {2, 3, 4, ...} with f(n) = n+1. f is a bijective function from IN onto {2, 3, 4, ...}. This proves that IN = {1, 2, 3, ...} is (Dedekind-)infinite (since {2, 3, 4, ...} is a proper subset of IN = {1, 2, 3, ...}).

Note: "In mathematics, a set A is (Dedekind-)infinite [...] if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A." (Wikipedia)

[WM]

Post by Me

Nobod[y] can prove that [IN] is [Dedekind-]infinite.

Liar!

I said: Nobod can prove that it is Cantor-infinite.

Post by Me
Just consider the function f: IN --> {2, 3, 4, ...} with f(n) = n+1. f is a bijective function from IN onto {2, 3, 4, ...}. This proves that IN = {1, 2, 3, ...} is (Dedekind-)infinite (since {2, 3, 4, ...} is a proper subset of IN = {1, 2, 3, ...}).

Interesting that you did not respond to Cantor-infinite. Of course you cannot. Nobody can. It is impossible to believe, without mental illness, that the infinite can be finished.

Regards, WM

[Me]

Post by WM
I said: Nobod can prove that it is Cantor-infinite.

Maybe Nobod can prove that. Could you quote a source where Nobod pubished his (or here) proof?

On the other hand, you still didn't provide a _proper definition_ (in the context of set theory) of the notion /Cantor-infinite/. So the relevance of Nobod's proof is not clear at all!

In the meanwhile I can show you a proof of the fact that IN is infinite:

Just consider the function f: IN --> {2, 3, 4, ...} with f(n) = n+1. f is a bijective function from IN onto {2, 3, 4, ...}. This proves that IN = {1, 2, 3, ...} is infinite (since {2, 3, 4, ...} is a proper subset of IN = {1, 2, 3, ...}). (Hint: The context of this proof is, as usual, set theory, of course.)

See: http://mathworld.wolfram.com/InfiniteSet.html

[WM]

Post by Me
Just consider the function f: IN --> {2, 3, 4, ...} with f(n) = n+1. f is a bijective function from IN onto {2, 3, 4, ...}. This proves that IN = {1, 2, 3, ...} is infinite

is Dedekind-infinite or potentially infinite. You do not prove exhaustability (= Cantor-infinity).

Regards, WM

[Ross A. Finlayson]

Post by WM

Post by Me
Just consider the function f: IN --> {2, 3, 4, ...} with f(n) = n+1. f is a bijective function from IN onto {2, 3, 4, ...}. This proves that IN = {1, 2, 3, ...} is infinite

is Dedekind-infinite or potentially infinite. You do not prove exhaustability (= Cantor-infinity).
Regards, WM

No, you just show it's the same as
1, 1,2, 1,2,3, 1,2,3,4, .... Then
where that wasn't already then it is
the exhaustibility with their union,
infinite, not just increment, next.

Simple.

Of course, then things are in terms
of that instead of the other.

Then, that it's also derivable the
other way for similar but different
conditions, often and usually the
point or infinity is axiomatic,
then cataloged under its properties.

For others it's derived and then
infinity is a natural feature of
the number(s).

What infinity's natural properties
get in the way of various extensions
of properties of models of numbers
and the numbers, here it goes
infinity's way, in some various
extensions of properties of models
of (eventually all the) numbers,
all the numbers.

Infinity built this way is having
its properties as it's built.

Set theory's quite in no way arbitrary,
except "everything's a set in set theory"
(as a theory with logical principles and
the constant 0 and omega, for example, in
a regular set theory, for example with all
the usual rules of inference).

Sets are fundamental and neat. Basically
sets and rules here then are as into functions.

Go "past" Cantor, not "away".

[Me]

<bla> Cantor-infinite <bla>

"Cantor-infinite"? Mind to define that term?

Maybe you meant /Dedekind-infinite/?

https://en.wikipedia.org/wiki/Dedekind-infinite_set

Or did you just mean /infinite/:

https://en.wikipedia.org/wiki/Infinite_set

or even /countable/:

https://en.wikipedia.org/wiki/Countable_set

[Jack Campin]

***@gmail.com wrote: [bollocks as usual]

Muck off, turd-for-brains.

-----------------------------------------------------------------------------
e m a i l : j a c k @ c a m p i n . m e . u k
Jack Campin, 11 Third Street, Newtongrange, Midlothian EH22 4PU, Scotland
mobile [phone] <http://www.campin.me.uk> Twitter: JackCampin

[c***@gmail.com]

Post by Simon Roberts
I apologize, I do not see the difficulty. Cantor's argument is based on an arbitrary set of infinite binary sequences. It only proves that there is an infinite set that cannot have a one-to-one correspondence with natural numbers. So?

That proof is based on the clumsy waste of "all" natural numbers.

First, there is no such thing as "all natural numbers" and a complete list.

Second, if assuming it counterfactually, then enumerate the same list by all powers of two. If there are diagonal numbers fabricated, enumerate them by all powers of 3. If there are further diagonal numbers fabricated, enumerate them by all powers of 5. And so on. You will never run out of natural numbers.

Post by Simon Roberts
My question would be, can you have an infinite subset of naturals that has a one to one correspondence with naturals.

Up to every number, yes. But every number belongs to a finite initial segment and is followed by (potentially) infinitely many others.

[g***@gmail.com]

Post by c***@gmail.com
*Every* entry of the Cantor-list differs from the antidiagonal. From this it is concluded that *all* entries differ from the antidiagonal and therefore the antidiagonal is not in the list.

NO!

any(row) digit(row,row) is num -> flip( digit(row,row) ) is num



SUM( i=1..row ) flip( digit(i,i) ) / 10^i

=/= any(row)

Post by c***@gmail.com
*Every* digit of an entry or of the antidiagonal is insufficient to determine a real number. From this it is not concluded that *all* digits of an entry or of the antidiagonal are insufficient to determine a real number.
*Every* rational number can be indexed by a natural number. From this it is concluded that *all* rational numbers can be indexed by natural numbers.
*Every* natural number leaves the overwhelming majority of rationals without index. From this it is not concluded that *all* natural numbers leave the overwhelming majority of rationals without index.

think of AXIOM OF EXTENSIONALITY



ANY(SET1) ANY(SET)

ALL(MEM) MEMeSET1<->MEMeSET2
<->
SET1=SET2




ALL MEMBERS FORM A *CONSTRUCTION*

BUT ALL SETS ARE JUST THE DOMAIN


SO THERE IS ONLY 1 QUANTIFIER AT WORK

[c***@gmail.com]

Post by g***@gmail.com

Post by c***@gmail.com
*Every* entry of the Cantor-list differs from the antidiagonal. From this it is concluded that *all* entries differ from the antidiagonal and therefore the antidiagonal is not in the list.

NO!

In set theory it is claimed.

Post by g***@gmail.com
any(row) digit(row,row) is num -> flip( digit(row,row) ) is num
SUM( i=1..row ) flip( digit(i,i) ) / 10^i
=/= any(row)

Post by c***@gmail.com
*Every* digit of an entry or of the antidiagonal is insufficient to determine a real number. From this it is not concluded that *all* digits of an entry or of the antidiagonal are insufficient to determine a real number.
*Every* rational number can be indexed by a natural number. From this it is concluded that *all* rational numbers can be indexed by natural numbers.
*Every* natural number leaves the overwhelming majority of rationals without index. From this it is not concluded that *all* natural numbers leave the overwhelming majority of rationals without index.

think of AXIOM OF EXTENSIONALITY

That does not change the fact that after every indexed rational number there follow infinitely many.

[g***@gmail.com]

Post by c***@gmail.com
That does not change the fact that after every indexed rational number there follow infinitely many.

thanks Santa!

[pirx42]

Post by c***@gmail.com
*Every* entry of the Cantor-list differs from the antidiagonal. From this it is concluded that *all* entries differ from the antidiagonal and therefore the antidiagonal is not in the list.
*Every* digit of an entry or of the antidiagonal is insufficient to determine a real number. From this it is not concluded that *all* digits of an entry or of the antidiagonal are insufficient to determine a real number.
*Every* rational number can be indexed by a natural number. From this it is concluded that *all* rational numbers can be indexed by natural numbers.
*Every* natural number leaves the overwhelming majority of rationals without index. From this it is not concluded that *all* natural numbers leave the overwhelming majority of rationals without index.

Is it you, Mücke?
After all these months we started missing you.

[Me]

*Every* [bla] *all* [bla]

Yeah, Wolfgang. Back on track? :-)

[Ross A. Finlayson]

Post by c***@gmail.com
*Every* entry of the Cantor-list differs from the antidiagonal. From this it is concluded that *all* entries differ from the antidiagonal and therefore the antidiagonal is not in the list.
*Every* digit of an entry or of the antidiagonal is insufficient to determine a real number. From this it is not concluded that *all* digits of an entry or of the antidiagonal are insufficient to determine a real number.
*Every* rational number can be indexed by a natural number. From this it is concluded that *all* rational numbers can be indexed by natural numbers.
*Every* natural number leaves the overwhelming majority of rationals without index. From this it is not concluded that *all* natural numbers leave the overwhelming majority of rationals without index.

The natural/unit equivalency function is a unique
counterexample to uncountability of the real numbers.

Or, "Cantor proves a line is drawn".

[Ross A. Finlayson]

Post by Ross A. Finlayson

Post by c***@gmail.com
*Every* entry of the Cantor-list differs from the antidiagonal. From this it is concluded that *all* entries differ from the antidiagonal and therefore the antidiagonal is not in the list.
*Every* digit of an entry or of the antidiagonal is insufficient to determine a real number. From this it is not concluded that *all* digits of an entry or of the antidiagonal are insufficient to determine a real number.
*Every* rational number can be indexed by a natural number. From this it is concluded that *all* rational numbers can be indexed by natural numbers.
*Every* natural number leaves the overwhelming majority of rationals without index. From this it is not concluded that *all* natural numbers leave the overwhelming majority of rationals without index.

The natural/unit equivalency function is a unique
counterexample to uncountability of the real numbers.
Or, "Cantor proves a line is drawn".

This is plain as the face.