I'm asking about the process that causes other mathematical beliefs to generalize to your beliefs about physical time in such fashion that physical time always seems to have the smallest model allowed by any of your mathematical beliefs. When I learn that the ordinal epsilon-0 corresponds to an ordering on unordered finitely branching trees, I don't conclude that a basket of apples is made out of little tiny unordered trees. What do the physical apples have to do with ordinals, after all?

Why, as you come to believe that Zermelo-Fraenkel set theory has a model, do you come to believe that physical time will never show you a moment when a machine checking for ZF-inconsistency proofs halts? Why shouldn't physical time just be a random model of PA instead, allowing it to have a time where ZF is proven inconsistent? Why do you transfer beliefs from one domain to the other - or what law makes them the same domain?

This isn't meant to be an unanswerable question, I suspect it's answerable, I'm asking if you have any particular ideas about the mechanics.