12
votes
1answer
834 views

What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory?

Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that's unclear, in this ...
15
votes
3answers
1k views

Any paradoxical theorems arising from large cardinal axioms?

If we accept the axiom of choice, we take the responsibility of living in a world in which, e.g., a ball in Euclidean 3-space is equiscindable to two isometric copies of itself (Banach-Tarski). So we ...