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Re: No Identity Bijection for Omega
Posted:
Sep 7, 2007 11:27 PM
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On Sep 7, 12:32 pm, george <gree...@cs.unc.edu> wrote:
> NO, one COULDN'T.
> Worse, one canNOT do THAT *even* in ZF+Infinity!
> The axiom of infinity says that AN infinite set containing all the
> naturals (finite ordinals) exists. It does NOT say that this set is
> omega.
OK, so you are using a slightly different definition of "omega"
than some of the other posters in this thread.
> The best you can do in ZFC(Including Infinity) is to approximate omega
> as the INTERSECTION of all the ordinals containing all the naturals.
> This means there will be a smallest-set-IN-YOUR-MODEL-of-ZFC
> containing all the naturals, and the naturals will be the only
> DEFINABLE
> elements of your model in this set. But this set in your model STILL
> might have MORE things in it than the naturals.
Obviously, you're referring to ZFC+IST, mentioned in other
Cantor threads, in which the set you mention above will
contain the nonstandard naturals. And indeed, one can't
prove the existence of a set containing only the standard
naturals, so it is indeed true that it is impossible in FOL
to specify exactly the set of finite naturals.
Thus you are defining "omega" to mean "the set of all
standard naturals." But I've seen omega defined differently
on other sites. We go back to metamath again:
http://us.metamath.org/mpegif/omex.html
This proves the existence of omega in ZFC -- indeed in
ZF, since the list of axioms used at the bottom of the
page omits Choice. Yet you just said that it's impossible
to prove the existence of omega in ZFC!
What's happening is that a different definition of omega
is being used.
http://us.metamath.org/mpegif/df-om.html
"Define the class of natural numbers, which are all ordinal
numbers that are less than every limit ordinal."
And of course, in ZFC+IST, the nonstandard naturals
_are_ ordinals that are less than every limit ordinal,
so they are elements of omega. As the posters in
that other thread "the shocking truth about the
natural numbers" would put it, the zero-place
symbol omega is mapped to a set other than the
set of all standard natural numbers in any model
of the theory ZFC+IST.
As I mentioned in the first metamath link, it is
provable in ZF that omega exists, and so it is
true in every model of ZF (including models of
ZFC and ZFC+IST) that the set to which
omega happens to be mapped in the model
actually exists.
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