Objections to the Use of the Axiom of Choice to Model the Physical World

Abstract

We look at platonistic mathematics and the application of this perspective in the physical world. We recognize paradoxes within Zermelo–Fraenkel set theory with the axiom of choice (ZFC) that conflict with physical reality, giving us reason to question if the axiom of choice should be so freely applied in theories of the physical world especially since it appears to enable a deterministic perspective. In theories of quantum physics, the axiom of choice is used to assume noncomputable numbers as initial conditions. This is equivalent to assuming a finite system contains an infinite amount of information at an instant in time; however, we know the world behaves probabilistically at the subatomic level. Our main purpose is to question the use of the axiom of choice when describing the physical world and advocate for alternate views of math, such as intuitionist mathematics, that avoid the assumption of determinism