Meta-metamathematics

[JOSEPH SHIPMAN]

Thesis: the most simply stated description of the main thrust of work in foundations of mathematics is “consider new axioms”.

Of course there’s other stuff we do, but if we never make any progress towards deciding, given two axioms whose conjunction is refutable, which to prefer, it’s not very satisfying.

I would like to ask something one level up: for what kinds of new axioms is it conceivable that some kind of persuasion could even occur?

Consider an alien civilization which used ZF + AD. They would regard some arithmetical statements as established that we don’t, such as consistency of Woodin cardinals.

What if they used ZFC + RVM? Then they’d get consistency of large cardinals below measurable.

On the other hand, if they used ZF + V=L, there aren’t any arithmetical statements, or even absolute statements, which they would have anything to say about that might persuade people to change their minds.

I propose the following:

“There is no physical experiment which could ever provide persuasive evidence for or against statements of set theory that are not absolute. Mathematicians who firmly disagree about such a statement, but who agree about fundamental physics, will never be able to settle the matter by doing computations or any other physical processes.”

If this proposition is correct, then arguing about, say, the continuum hypothesis can be viewed as a “religious dispute”. Even though CH is refuted by ZFC+RVM, CH-believers will not have to regard, say, physical evidence for the consistency of cardinals below a measurable as evidence toward RVM, because direct evidence for CH itself will not be something that correct physical theories can provide. Attention should be focused on axioms with new arithmetical (or at least absolute) consequences.

What is your opinion of this proposition? I especially want to hear from physicists.

— JS 

Sent from my iPhone