[FOM] Platonistic philosophy of Mathematics (response to Budnik's post)

[Tom Dunion]

On October 11, Paul Budnik said:

>...Cantor's proof that the reals are not
>countable establishes the incompleteness of definability in any formal
>system. One can always use diagonalization to define more reals. Large
>cardinal axioms can be thought of as proposed extensions of this idea.
>They postulate hierarchies of yet to be defined reals, functions on
>reals , etc., that /apply to any "correct" formal system/ /that can be
>defined/. Statements about uncountable sets do not necessarily have a
>definite truth value since the sets they refer to cannot have a definite
>meaning in this universe because there is no rigorous way to define the
>absolutely uncountable.

This calls to mind a penetrating question of Feferman's some years back,
"What are the axioms of ZFC axioms *for*?

One tempting approach might be to analogize to the axioms characterizing
those properties all groups share, and say for example, ZFC is the axiom
system for something we might call "universe" theory (i.e. models of ZFC).
As a pedagogical tool, this is very helpful in Tim Chow's well-recommended
article, "A beginner's guide to forcing" (from which I borrow the idea).

However, Zermelo surely was not thinking of various universes which all just
happened to satisfy his proposed axioms.  Enamored of Cantor's bold leap
into a conceptual Platonic world where completed, actual infinities resided,
and stung by criticism of his proof of the well-ordering theorem, he developed
his system to make set theory impervious to challenge or self-contradiction.

The fact that (as Prof. Budnik points out) the axioms of ZFC can point to a
variety of as yet to be defined correct formal systems, may be telling
us that ZFC,
as much as it is such an amazing "gold standard" for formalization of so much
of Mathematics -- is however insufficient to characterize the fullness of that
Platonic realm that Cantor discovered.   Why should appreciation of
transcendent
truths be left to mystics and philosophers who have perhaps but dimly perceived
some things which may be explored also by those who care about Foundations?

If large cardinals be admitted into our catechism -- and I am not saying they
ought not be -- why not celebrate also the mystic side of "Saint Goedel"?

TD