Meta-metamathematics

[JOSEPH SHIPMAN]

Define “the”.

You talk about the consistency of “the” quantum-mechanical standard model. What, exactly, does that mean? Can you formalize it?

All the theoretical physics people do can, apparently, be formulated in weaker systems such as Zermelo set theory. But that doesn’t necessarily show its consistency, because there may be algorithms to calculate experimental predictions which are assumed to halt but which don’t. 

In my thesis I showed that certain attempts by the physicists Itamar Pitowsky and Stanley Gudder to make quantum mechanics deterministic and local, getting around the Bell Inequality by the trick of having Fubini’s theorem fail for physically relevant but necessarily nonmeasurable functions so that measuring observables in different orders corresponded to iterating integrals in different orders, did in fact need to go beyond ZFC (they assumed CH), because it was consistent that the Fubini property held for all functions where it wasn’t trivially refutable. So the consistency of their models wasn’t a new axiom (because CH is consistent), but the models themselves included a new axiom, necessarily.

— JS

Sent from my iPhone

> On Jan 11, 2022, at 3:50 PM, Hendrik Boom <hendrik at topoi.pooq.com> wrote:
> 
> On Thu, Jan 06, 2022 at 01:46:45AM -0500, JOSEPH SHIPMAN wrote:
> 
>> 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.
> 
> Physicists do appear to act as if the consistency of the Quantum-mechanical
> Standard  Model were an axiom.
> 
> This isn't the continuum hypothesis, but is it 
> the kind of mathematical axiom you are talking about?
> 
> Or is it instead part of the physics?
> 
> -- hendrik