Is applied statistical mathematics also can be explained by the zfc set theory?

Question

I know that here there is already a lot of explanations about the zfc/zf/aca axioms but i wanted to ask if hypothetically people realy wanted to explain applied statistics to a creature that only understand axioms and logic derived from them will it be posible (even in a very complicated and uncomfortable way) ? And if so will it work with ather applied math subjects like mathematical chemistry?

Answer 1

Of course there is no way to prove this without actually doing it, because "applied statistical mechanics" is not a formal term, but everyone's intuition who has studied ZFC will be that yes, it is possible. ZFC is a very powerful theory, that allows you to "encode" mathematical objects of almost unnecessary complexity.

In particular, it is standard to formalize probability theory in ZFC set theory.

Answer 2

I think your question has some deep implications after all. All objects of ZFC are sets, even the numbers are represented as sets. Let's call these sets pure sets, all of their members are sets. I was wondering that in everyday mathematics we do sometimes use non-pure sets.

For example, even in the most formal measure-theoretic treatment of statistics, we define a probability space, that is a triple of the sample space, the sigma-algebra of events, and a probability measure. The elements of the sample space are outcomes, e.g. coin flip is H / T. I don't think in this sense the sample space is a pure set. So if we dig in deep, the answer might be no after all.

The only problem is that outcomes are not a pure set usually in the applications of statistics. (But do not worry, there are versions of ZFC with atoms shown to be equiconsistent with ZFC, so this could be a way to go about formalizing statistics in a ZFC setting.) On a personal note, I believe no matter how deep we go with our axioms, there will always be some bridge-principle needed that explains why mathematics can be used effectively in our models of reality. Without the metaphysical bridge / trust / (even implicit) belief / explanation, that provides a connection to the real world, mathematics is useless (but beautiful).

On most days, however, using maths as-is without digging this deep seems to be sufficient and quite effective in my experience.