What are the base assumptions we make in mathematics?

Question

In any proof, we establish (or attempt to establish) a fact based on previously established facts. Each of these previously established facts in turn must have been established using a set of then pre-established facts, and so on and so forth until either:

An infinite sequence of regressive proofs is reached, which is unlikely, as that would require an infinite body of mathematics just to have reached the point we are at now, considering every fact now by definition above would be built upon this infinite stack of facts, or...
We reach our base facts which are assumed or used as defining math itself, upon the foundation of which all other facts are proven

So what are these base facts underlying mathematics? Are they defined somewhere?

In addition to the axioms of set theory, we usually assume some basic logic, which is essential to allow us to write proofs in the first place. But the key thing is "axioms," and most of the ones we assume are discussed in set theory. I've found Halmos's book "Naive Set Theory" to be a fine place to start learning about these most basic axioms. — John Hughes, Nov 20 '18 at 15:40
You may be interested in the question Founding mathematics from a set of axioms.. — Mark S., Aug 18 '20 at 17:29

score 3 · Answer 1 · answered Nov 20 '18 at 15:34

3

Usually, Zermelo–Fraenkel axioms.

answered Nov 20 '18 at 15:34

Federico

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What are the base assumptions we make in mathematics?

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