[FOM] A proof that ZFC has no any omega-models

[Monroe Eskew]

On Feb 24, 2013, at 3:12 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

> O.K., so in your scenario nothing really changes.  Is that supposed to be a bad thing?
> 
> I mean, the whole point of Hilbert's program, as I understood it, was to enable us to stay in Cantor's paradise while easing ontological concerns that the "government" might have.  Although we now know that Hilbert's program can't succeed in its original form, it seems to me that you would have challenged Hilbert at the outset, asking him what the point of his program was all about if it wasn't going to lead to any change in mathematical practice.  But surely it's clear that the whole point of the program was to find good reasons *not* to change mathematical practice.

My understanding is that Hilbert's concerns were not about metaphysics but about consistency and the epistemology thereof.  In response to the paradoxes, he hoped to prove consistency from the bottom-up.  If something like this were possible and were discovered, we might imagine that it would have a profound effect mathematical method.  Furthermore, the failure of Hilbert's program led to very important mathematical developments, new fields, and new practices.

On the other hand, purely ontological concerns don't seem to lead to anything interesting.  If we can just add simple prefixes and suffixes to mathematical work in a uniform way to satisfy the ontologically conservative, then it seems we've found a rather boring solution to their problems.  And of course this is something anyone can do on their own: Have a metaphysical concern about some piece of mathematics?  Then if it suits you, imagine starting with a statement of the formal assumptions, and then when you're done reading, close off the conditional proof and draw your conclusion as to what those assumptions prove.  If the "government" had its way in my scenario, then it might look to most people that they weren't concerned with the meat of what the mathematicians were doing, just that they weren't doing it with good manners.

My overall point is that if the only concerns about foundations of math are ontological, then this is not very interesting.  But if we move beyond ontology to epistemology and methodology, there are many interesting questions, like why should we think it is reasonable to assume the consistency of ZFC + there exists a measurable cardinal?  We can't just say because no one has found a contradiction in many years, because that applies to too many things, like the Riemann Hypothesis and its negation.  More generally, whether we adorn them with ontologically deflationary ornaments or not, what kinds of statements should we count as legitimate starting points for proofs?  Standard large cardinals?  Determinacy?  The omega conjecture?  Saturated ideals?  Unique branches hypothesis?  PCF conjecture?  Perhaps eschew set theory and work from a category theoretic framework?  This type of question seems to me like a serious inquiry about foundations of mathematics, whereas purely metaphysical debates just seem like quibbling.

Best,
Monroe