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For example Alexander Grothendieck did not care much about the formal logical foundation of his work. So, I believe that this would make it highly complicated
by wolfgke 7y ago
For example Alexander Grothendieck did not care much about the formal logical foundation of his work.
So, I believe that this would make it highly complicated to even formulate some very new math at the borderlands of our knowledge.
- Koshkin 7y agoWell Category Theory is a pretty solid logical foundation...
- wolfgke 7y agoCategory 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...