For what I know the answer seems to be "It can't handle them" because the axiomatic set theory (at the least in its classical variation) allows only one type of objects: sets. So, for example, we can't have set of people because its elements wouldn't be just sets, they would be individual human beings (Like Donald Trump). Theoretically we could try to workaround this by creating subset "of all Donald Trumps who are current presidents of the USA". But it wouldn't be a real solution because such set must still contain Donald Trump, and as we all know Donald Trump isn't a set. Thus axiomatic set theory would be inadequate for such application.
2026-04-24 08:09:37.1777018177
How can axiomatic set theory handle real-life objects?
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Since the particles in the universe are countable, you may assign an integer to each one. There are well known ways to model integers as sets. Now take the system of particles that currently comprise Donald Trump. This is a subset of the integers so now you have a representation of Donald Trump as a set of sets.
Edit: A major objection to my answer is that my model is incorrect in light of modern and possibly future theories of physics. One could wonder if set theory is enough to model metaphysically correct theory of physics. I will remain agnostic to that question, but I would challenge anyone who claims to know for sure that the answer is in the negative.