Question

How would math be different under the Solovay Model instead of ZFC?

Answer 1

If your question is really how would mathematics be different, the answer is ­— **it wouldn't**. Mathematics does not change based on which foundational theory gets chosen. It's the other way around, i.e., something is considered a foundational theory if it's able to express all of the mathematics we are interested in anyway.

Also, Solovay model is not a theory. It is a particular model of ZF showing the relative consistency of some statements independent from the axioms.

You are comparing apples and oranges, i.e., an extension of a theory (ZFC as an extension of ZF) with a particular model of that theory (Solovay model of ZF). so, the question about which one should be preferred is moot, we're talking about two different kinds of things.

If you insist on comparing the two, I would advise that you do not think of either ZFC or Solovay model as being the foundation of mathematics. Take ZF as the foundation and see both ZFC and Solovay as certain "specializations" of ZF, the former obtained by adding an extra axiom, and the latter literally being a particular realization (a model) of ZF.

Answer 2

Since my post more of a general question, I'll give some context. Zermelo-Frankel Set Theory with Choice is the most widely used axiomatic foundation of mathematics. The Solovay Model is an alternative axiomatic foundation. The Solovay model has all the same axioms as Zermelo-Frankel Set Theory, but with the additional axioms that there exists an inaccessible cardinal and that the axiom of choice is instead the axiom of countable choice so all countable collections of sets have a choice function (instead of the standard axiom which is that all collections of sets have a choice function). The first interesting thing about the Solovay Model is that all subsets of the real numbers are measurable, but I'm curious as to what other implications this would have and why it isn't strictly preferrable to ZFC. The only other implication I know of is that only vector spaces of a countable dimension have a basis.