What is the most basic set of axioms that one can use in proofs? As in, the axioms are irreducible. The most basic set of irrefutable rules in mathematics. I assume it has something to do with number theory, but what are they?
The axiomatic method essentially starts with a set of axioms (not further analyzed), and develops their consecuences. I.e., there are the group axioms, from which a rich theory is developed. There is a (totally separate) axiomatization of (Euclidean) geometry,Peano's axiomatization of arithmetic, and even Blum's axioms for algorithmic complexity. There aren't "fundamental axioms for all mathematics" in the sense Euclid envisioned. The axiomatic idea spilled over into other areas, you can take Newton's laws as axioms for (classical) mechanics, the laws of thermodinamics are axioms for that area of physics.