Incompleteness versus a Platonic multiverse

The Platonic multiverse view is that there are multiple and incompatible concepts of a set with corresponding Platonic universes. For example the continuum hypothesis may be true in some of these universes and false in others. This philosophical view leads to an approach for exploring mathematics that is similar to an approach that stems from the more conservative view that infinity is a potential that can never br fully realized. My version of this view sees infinite collections as human conceptual creations that can have a definite meaning even if they cannot exist physically. The integers and recursively enumerable sets are examples. In this view infinite sets are definite things only if they are logically determined by events that could happen in an always finite but potentially infinite universe with recursive laws of physics. This includes much of generalized recursion theory, but can never include absolutely uncountable sets. “Logically determined” is a philosophical principal that can be partially defined rigorously, but will always be expandable. In this view uncountable sets can be definite things only relative to specific countable (as seen from the outside) models. Just as Gödel proved that any formal system embedding basic arithmetic must be incomplete in provability, Cantor’s uncountability proof plus the Löwenheim-Skolem theorem prove that any sufficiently strong formal first order system must be incomplete in definability. One can always define more reals. In this philosophical view uncountable sets are guides to how mathematics can be expanded. Thus at different stages or paths of development one might assume different and conflicting axioms about