I'm new to studying reverse math, how strong is the statement of the existence of a nontrivial ultrafilter on $\omega$? Does it have the same strength as one of the big 5 subsystems, or does it lie outside of them?
Here's another asked questions that was similar, but seemingly from a different perspective. https://mathoverflow.net/questions/258310/how-strong-is-the-existence-of-a-non-trivial-ultrafilter-on-omega
Thanks for any help in advance.
Reverse mathematics is the wrong tool for this: it takes place in the context of second-order arithmetic (= natural numbers and sets of natural numbers) and so the existence of an ultrafilter (= a set of sets of natural numbers with certain properties) can't even be stated directly in the language.
Expanding reverse math to broader languages is the task of higher reverse mathematics. In this context, Henry Towsner showed that the existence of an ultrafilter is conservative over the theory ACA$_0$ (and a number of other facts).