[FOM] Foundational Challenge

[Joe Shipman]

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 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
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).

— JS

Sent from my iPhone