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Added: I believe that the current proof we have of Fermat's Last Theorem uses the existence of a(t least one) Grothendieck universe. However, my understanding is that this dependence can be completely removed due to the fact that Fermat's Last Theorem has small quantifier complexity. I imagine that proofs of statements with higher quantifier complexity that use Grothendieck universes, do not necessarily have a way of removing their dependence on said universes. How would we tell if such arithmetic statements are true of the natural numbers?

Added: I believe that the current proof we have of Fermat's Last Theorem uses the existence of a(t least one) Grothendieck universe. However, my understanding is that this dependence can be completely removed due to the fact that Fermat's Last Theorem has small quantifier complexity. I imagine that proofs of statements with higher quantifier complexity that use Grothendieck universes, do not necessarily have a way of removing their dependence on said universes. How would we tell if such arithmetic statements are true of the natural numbers? [Some of this was incorrect, as pointed out by David Roberts and Timothy Chow.]

2nd addition: There are some theories that we believe prove false arithmetic statements. Assuming the natural numbers can consistently exist (which we do!), then both PA+Con(PA) and PA+$\neg$Con(PA) are consistent, but the second theory proves the false arithmetic sentence $\neg$Con(PA).

My question then might be rephrased as:

What principles lead us to believe that "Universes" is a safe assumption, whereas "$\neg$Con(PA)" is not safe, regarding what we believe is "true" arithmetic? (Next, repeat this question regarding the axiom of power set.) Is any theory that interprets PA "safe", as long as it is consistent with PA, and PA+Con(PA), and any such natural extension of these ideas?

Another way of putting this might be as follows:

Is the assumption Con(PA) a philosophical one, and not a mathematical one?

This ties into my previous question that I linked to, about describing the "real" natural numbers.