Re: Ultimate debunking of Cantor's Theory

Peter Webb wrote:


"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?


This may be true. After all, in ZF, we can still carry out induction through the ordinals. A natural number can be defined without Infinity, and then we can make statements like "For all x, if x is a natural number, then..." In fact, can't we just use transfinite induction to prove such statements?

However, it seems that without Infinity, we wouldn't be able to establish the existence of negative numbers, let alone rationals and reals. We could still define them and make statements about them, I think, but we would have no guarantee that the statements are not vacuous.

But I am no expert. Experts care to chime in?


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.


--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan

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