On Friday, July 21, 2017 at 10:13:35 PM UTC-4,
Post by Jim BurnsOn Friday, July 21, 2017 at 9:31:47 PM UTC-4,
On Friday, July 21, 2017 at 6:04:12 PM UTC-7,
Post by Dan ChristensenGiven the real numbers, what axioms of ZFC would you
use to extend them to include an infinite number?
Axiom of Union
The union of what set(s)?
Using one-sided Dedekind cuts,
the union of the set of real numbers is just
the set of rational numbers.
Yes, the set of rational numbers is infinite,
That's a different kind of infinity, cardinality.
but it would be a stretch to call it the extension
of the reals to "infinity."
Recall that you gave the real numbers.
For each real number x (of the unextended reals),
there is exactly one set of real numbers X = {y| y < x}
the set of reals y less than x.
Define an injection L: R -> P(R) so that
L(x) = X = { y | y < x }
Notice that
x =< y <-> L(x) sub L(y)
We can extend the image of L in R (the Dedekind cuts
in R) with the sets R and {}.
(Remember, L(R) doesn't have reals,
it has sets of reals.)
Let's call L(R) U { R, {} } = L(R)+
How do you go from this set to the notion of
some kind of "infinite number?"
And that it might somehow be equal to 1/0?
I'm going to assume that you've read all the way to
the end of my post. An important of the answer to
your question is "Those things I said next".
I'm assuming that we already have the usual set of real
numbers with their usual operations, and that we have
ZFC too.
What I'm doing is extending those usual real numbers with
a positive infinite number _larger than all_ usual, finite
real numbers and a negative infinite number _less than all_
usual, finite real numbers. It's certainly not the only way
to extend the reals to infinite numbers, but it's one way.
I'm not sure if you are concerned that the infinite
numbers I named +inf and -inf exist. Supposing you
are concerned, consider that the sets R and {}
that correspond to them do exist (by assumption, given
the reals and ZFC) and there is a bijection between R+ and
L(R)+ . Anything non-self-contradictory that we can say
about the elements of L(R)+ we should be able to translate
(using L(x)) into a statement about the elements of R+ .
As far as +inf and -inf are concerned, they exist because
(i) I said so, and (ii) they do not create contradictions
(no more than the reals or ZFC do). These are the same
reasons we say 0, 1, 2, ... exist.
It's not enough just to say that +inf and -inf exist.
Do they play well with others? How do the extended versions
of <, +, -, *, / work?
One thing I assumed is that we want the _usual_ part of
the extended operations to look like the _unextended_
operations. That bypasses deep discussions of whether
the finite, extended reals are "really" the reals. This
would be my great preference, to avoid that discussion.
I already showed how to extend =< by placing +inf at
the upper end of the extended reals and -inf at the
lower end of the extended reals.
If that was all I needed to do, there would have been
little point to introducing Dedekind cuts. But we can
define the extended +, -, *, / by looking at the
operation on finite unextended reals in those cuts.
This is what's done in the usual construction of the
real numbers from Dedekind cuts of the rationals.
By the way, _I won't know_ until I look whether this will
give us 1/0. I'm taking "larger (smaller) than all the
finite numbers" to be the defining characteristic of
+inf (-inf) and then seeing what follows from that.
(This is not the only way to do things.)
Notation:
[x] = { y e R | y < x }
Define an "addition" operation [+] on L(R)+
[x] [+] [y] = [z] <->
[z] = { w e R | (Eu e [x])(Ev e [y]( u + v = w ) }
where the '+' inside the definition of [z] is the usual
addition on the reals.
From this definition, we can see that, for all
_finite_ x and y
[x] [+] [y] = [x + y]
which is precisely what we want.
We also have, from the same definition,
[x] [+] [+inf]
= [x] [+] R
= R
= [+inf]
Therefore, it seems reasonable to define, for R+, x finite,
x + +inf = +inf
in order to preserve, for infinite numbers,
[x] [+] [y] = [x + y]
By similar arguments,
x + -inf = -inf
+inf + -inf = -inf
(This last one surprised me.)
We can try extending the definitions for
neg([x]) = [-x]
inv([x]) = [1/x]
[x][*][y] = [x*y]
as we extended
[x][+][y] = [x + y]
It might or might not give us something for 1/0.
Post by Jim BurnsNotice that
all x in L(R)+, x sub R
and
all x in L(R)+, {} sub x
If we extend R to R+ with two constants +inf, -inf
We could extend our infection L(x) to L'(x) so that
L': R+ -> L(R)+
with
L(+inf) = R
and
L(-inf) = {}
Then we would still have
x < y <-> L'(x) sub L'(y)
only now with two more elements such that
all x in R+, x =< +inf
and
all x in R+, -inf =< x
If I'm wrong, I hope someone corrects me, but that
looks like the reals extended with positive and negative
infinity.