Philosophers sometimes ask which axioms are "true". Of course, no one believes that there is a serious question as to whether the axiom of commutativity for groups, or the Parallel Postulate, is true. In the former case, we're just talking about the class of models of the axioms. In the latter, we talk about a different geometrical space by giving a different answer to the Parallel Postulate question. But philosophers worry about the axioms of set theory and arithmetic, as if those were different. In general, I don't see the motivation for this. I don't see why we should regard a "universe" satisfying Determinacy differently from geometrical space satisfying the negation of the Parallel Postulate. But it does seem hard to hold this view across the board. After all, the view would be something like "every (first-order) consistent set of axioms is equally legitimate", and the claim that a theory is consistent is a Pi_1 arithmetic sentence which one can consistently deny! So, assuming that there is an objective fact (not sure how to put it) as to whether at theory like PA is consistent, is the most radical but coherent view in the neighborhood that "all Pi_1 sound theories are equally legitimate"? How do you think about this issue?
Following Carl's suggestion (below), my concrete question is the following. Is there anything subtly inconsistent in the idea that PA + Con(PA) stands to Euclidean geometry as PA + ~Con(PA) stands to Hyperbolic Geometry -- equally true, simply true of different structures?