3 ms·
Category theory depends on set theory. Unluckily, formulating a category of all sets/groups/rings/... is not easy because there exists no "set of all sets/group
by wolfgke 7y ago
Category theory depends on set theory. Unluckily, formulating a category of all sets/groups/rings/... is not easy because there exists no "set of all sets/groups/rings/...".
One way to solve this problem is to replace ZFC by its non-conservative extension Tarski–Grothendieck set theory
https://en.wikipedia.org/w/index.php?title=Tarski%E2%80%93Grothendieck_set_theory&oldid=918247704 https://en.wikipedia.org/w/index.php?title=Tarski%E2%80%93Gr...
to ensure that a Grothendieck universe
https://en.wikipedia.org/w/index.php?title=Grothendieck_universe&oldid=899048612 https://en.wikipedia.org/w/index.php?title=Grothendieck_univ...
always exists.
- QuesnayJr 7y agoMost of category theory as used in ordinary mathematics can be formulated in ZFC. The category of all groups, etc. is just a propositional formula. If you prefer to explicitly talk about classes, then you can go with NBG, which is a conservative extension of ZFC. Grothendieck needed an extra axiom because he wanted to consider large Grothendieck topologies. The consensus seems to be that universes are avoidable: https://mathoverflow.net/questions/35746/inaccessible-cardinals-and-andrew-wiless-proof https://mathoverflow.net/questions/35746/inaccessible-cardin...