Objective mathematics in a finite unbounded universe

Most mathematicians think that first order arithmetic is objectively true in some sense. Stronger formal systems lead to increasing skepticism. Substituting the philosophical question of which mathematical statements are objective with the question of which mathematical statements are logically determined by events that could in theory occur in the physical universe, as we understand it, provides a partially mathematical definition of one form of objectivity. Because of incompleteness any correct mathematical definition of ‘logically determined’ can be expanded. Much countable mathematics including the minimal standard model for set theory may meet this definition of objective. Cantor’s uncountability proof and the Löwenheim-Skolem theorem prove that any consistent sufficiently strong first order theory can be expanded with more reals. The absolutely uncountable cardinal hierarchies cannot meet this definition of objective, but they implicitly define tools for the expansion of objective mathematics. Because objective mathematics is about what may be meaningful in the physical universe, it suggests techniques for constructing partial computer models of mathematical universes that can then be explored experimentally to help develop mathematical understanding and intuition.