How it makes sense to talk about consistency of theories?

Question

So I was reading this post Probability axioms (Kolmogorov) and so in the answer there was said that it doesn't make sense to talk about consistency of probability axioms because it is not a formal system but just a part of measure theory, but now, can't we say in the same way that it doesn't make sense to talk about consistency of PA, Real numbers, Geometry if they all can be reduced to ZFC ? What makes probability case different from other theories ?

Answer 1

I'm with Andres on this one: I disagree with the linked answer and Asaf's comment.

Let me respond to Asaf's comment first. While "the free group on two generators is consistent" doesn't make any sense, there is an obvious set of axioms for "the free group on two generators" - namely, the axioms of group theory together with the negation of every nontrivial identity involving two new constant symbols - and we can ask whether that set of axioms is consistent; the most natural proof that it is, of course, consists of first proving the soundness theorem and then constructing the free group on two generators and showing it satisfies the above-mentioned theory. This argument requires some "metamathematical overhead" - we need to be able to talk about groups, axioms, and satisfaction - but for example is easily performable in ZFC.

Now on to the probability space example. Unlike the above situation, the axioms for probability spaces are not first-order (e.g. even saying that something is a $\sigma$-algebra takes us well outside of FOL). That's not inherently a problem, but it does mean that we have to be working in a metatheory which can make sense of the relevant logic (second-order logic will do the job just fine). ZFC, for example, will more than suffice: ZFC proves that the axioms for probability spaces are satisfiable.

Note that I said "satisfiable" rather than "consistent." Generalized logics don't necessarily come with sound and complete proof systems; for example, in a very strong sense there is no good proof system for second-order logic. When we work with first-order logic, consistency is a perfectly meaningful notion, but when handling more complicated logics the semantic side of things is much better than the syntactic side of things. However, one immediate corollary of the above-mentioned fact is that ZFC proves "For every proof system which is sound with respect to second-order logic, the axioms of probability spaces are consistent with respect to that proof system." Note that this really is trivial (since soundness exactly tells us that no new entailments can be produced), so despite the above concerns there is a very good sense in which ZFC proves "the probability space axioms are consistent."