Re: set theories

On Thu, 4 May 2006, Mark-T wrote:

I've been doing some research into set theory, and I have
a question which none of the literature addresses: we have

Research? Naw, don't you mean studies?

ZFC and Peano's axioms, and that's it. Where do they come
from? I mean, is there some objective criteria which they satisfy,
and nothing else does? Or is it a matter of aesthetics -
"these theories seem pretty nice, why bother thinking up new ones"

They come from Zermelo, Frankel and Peano. No, they are not categorical.
It was a matter of a lot of sweat from a lot of mathematicians.

Also, is PA a subset of ZFC, insofar as the provable theorems?
If not, what is the relationship between them?

No, they are different theories. You can with the axiom of infinite,
construct sets in ZFC and a successor operator, that fulfill the axioms of
PA. No, subset is absurd. Embed is more appropriate.
.

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