The axiom of choice applies exclusively to infinite sets, and the finite restriction is a consequence of ZF without AC. Since we cannot actually construct infinite sets, infinitesimals, or even irrational numbers, the consequences of ZFC over and above ZF in the real world are negligible at most, and almost certainly nonexistent.
ZF is useful because its language maps very well onto the real world, but as an example unifying general relativity and quantum mechanics has been difficult.
These are not related statements. There are a lot of ways to choose axioms, but I’m not aware of any of them having any consequences in regimes applicable to physics. Any choice of axioms meant to describe the world has to support some basic conclusions like the validity of arithmetic, and that puts great restrictions on what the axioms are, and what they could possibly say about a physical theory.
As I understand it geometry was axiomatic for much longer, and the discovery of non-Euclidean geometries required separating the original axioms for different topologies.
Geometry was nominally axiomatic for many centuries, but not rigorously axiomatic until the study and development of non-Euclidean geometry, which was the beginning of the rigorous axiomatization of mathematics. Prior to that, statements were frequently taken as axioms on purely intuitionist grounds (in many subfields); it took the demonstration that Euclid’s fifth axiom (the parallel postulate) was unnecessary to make mathematicians actually check their assumptions and establish sets of axioms which were consistent.
The axiom of choice applies exclusively to infinite sets, and the finite restriction is a consequence of ZF without AC. Since we cannot actually construct infinite sets, infinitesimals, or even irrational numbers, the consequences of ZFC over and above ZF in the real world are negligible at most, and almost certainly nonexistent.
These are not related statements. There are a lot of ways to choose axioms, but I’m not aware of any of them having any consequences in regimes applicable to physics. Any choice of axioms meant to describe the world has to support some basic conclusions like the validity of arithmetic, and that puts great restrictions on what the axioms are, and what they could possibly say about a physical theory.
Geometry was nominally axiomatic for many centuries, but not rigorously axiomatic until the study and development of non-Euclidean geometry, which was the beginning of the rigorous axiomatization of mathematics. Prior to that, statements were frequently taken as axioms on purely intuitionist grounds (in many subfields); it took the demonstration that Euclid’s fifth axiom (the parallel postulate) was unnecessary to make mathematicians actually check their assumptions and establish sets of axioms which were consistent.