Re: Skolem's Paradox and why is math the way it is?

From: J.E. (troubled6man_at_yahoo.com)
Date: 09/30/04 

Date: 30 Sep 2004 08:25:40 -0700

> > what I say wrong, independant of my spelling, that's why I was using
> > caps because people seemed to think I meant "set" when I said "class"
> > and I wanted people to be able to easily go back and read clearly what
> > I tried to say before they responded to things I didn't say. I'm very
> > frustrated that either some people here aren't trying to understand me
> > or that I'm not being clear because people keep argueing against straw
> > men and not against me. Which is sad because I don't care about these
> > straw men, but I'd love very much to have some succeed at explaining
> > how I'm wrong, or just telling me why the ZF axioms are good.
>
> Ah yes, I suddenly realized you use the code "ZF" and don't know at all
> what it means. For instance, surprise, there are no mention of R (or its
> non-denumerability) in them... As for the reason they are good : 1) they are
> simple. 2) in your sense, they are not fixable : *any* (consistent) axiom
> set has denumerable models (Skolem theorem). And (surprise) ; those axioms
> *don't* speak of models either.

Is there not a theorem that says that the set generated by feeding the
output of the axiom of infinity to the input of the axiom of powers
has no bijection to the set generated by the axiom of powers? I
thought that the reason that the theorem held was that for any axioms
you make, there could be a countable model with all the sets that are
provable for just those axioms, but that there can always be other
sets in other models that are based on axioms that *could* have been
consistantly added to the ZF axioms. Do you see how I'm using model
theory to interpret the theorem, and that this interpretation is
different than the usual interpretation that says that there is
actually lots and lost of subsets of the infinite set? I'm beginning
to think one has to use second order things to actually force all
models to be bigger than the infinite set.

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