[FOM] Foundational Challenge

[Paul Blain Levy]

Dear Joe,
> Date: Thu, 9 Jan 2020 10:45:44 -0500
> From: Joe Shipman <joeshipman at aol.com>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: [FOM] Foundational Challenge
> Message-ID: <02F5CC89-6953-4324-9CCF-AA8F8662503A at aol.com>
> Content-Type: text/plain; charset=utf-8
>
> ?When I read about ?alternative foundations?, I feel a bit like the physicists who wade through many papers on ?Interpretations of Quantum Mechanics? and ?Alternative Theories? without ever getting to either an honest new prediction of an experimental result that the standard theories can?t predict, or insights allowing standard predictions to be derived significantly more easily.
>
> Therefore I issue the following challenge to proponents of any kind of ?alternative foundations? to ?ZFC plus large cardinals?.
I am such a proponent, so let me examine your challenges.
> I CHALLENGE YOU TO IDENTIFY:
>
> An EXAMPLE of mathematics done in any ?alternative foundation? to set theory, where EITHER
> a) it is not obvious that any result derived that is statable in set theory will be provable in ZFC plus a large cardinal of some kind

This is a good challenge to put to someone who criticizes ZFC for being 
too weak.  But not to someone like me, who criticizes it for being too 
strong.

> OR
> b) it is an ARITHMETICAL result with a significantly shorter formal derivation from first principles than is possible in ZFC (?significantly? means ?more than the usual amount of work to translate proofs formalized in Second Order Arithmetic into ZFC proofs?, which is a well-understood quantity).

Not relevant to my criticism of ZFC, which has nothing to do with proof 
length.

Paul