Re: Ultimate debunking of Cantor's Theory


"G. Frege" <nomail@invalid> wrote in message news:d3sd93h003qnh3pjt16qq743d1u67mssh0@xxxxxxxxxx
On Fri, 13 Jul 2007 13:23:46 +1000, "Peter Webb"
<webbfamily@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:


ZF would work perfectly well for many purposes without the Axiom
of Infinity; it is included because without the Axiom of Infinity,
we can't talk about infinite sets at all, and that is so boring
that it was "bundled into" the core axioms of ZF.

Moreover a "set theory" could hardly serve as a ("the") mathematical
foundation, if we could not even derive the Peano Axioms from them.
The other way round: we can derive virtually all mathematics (or at
least a good deal of it) from ZFC.


Whilst I agree with you, it is my suspicion that you don't need the Axiom of Infinity for the Peano axioms. The only place you come close is in the induction axiom, which can (I suspect) be re-written to avoid the concept of "all numbers".

You probably know if this is true or not - you don't need the Axiom of Infinity for PA, just as we don't need the AxC for PA either?

The Axiom of Infinity seems to me to be a lot like the Axiom of Choice - very useful in set theory, but not completely self evident, and not necessary for the construction of numbers.


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