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$\begingroup$ The "FOL + categories" example is helpful, thank you. It makes intuitive sense to not add something to the core that could easily be supported outside it. $\endgroup$
– Chris Henson
Commented May 8 at 10:42
$\begingroup$ I don’t understand your first two paragraphs. If you replace “category” with “set”, wouldn’t you be saying that Isabelle/ZFC, Metamath/set.mm, and Mizar are just a core of FOL? I’m not sure what “core” means in this case? They are all “FOL + sets”. Maybe it comes down to if OP is looking for a system to do category theory or a system to do all of mathematics but use category theory as the foundation. I assume it is the second. I’m not sure the second one is a “good” approach, but I think it is feasible, no? And “FOL + categories” is one way to do it. Is see no reason to rule it out. $\endgroup$
– Jason Rute
Commented May 8 at 10:51
2

$\begingroup$ If you have bare FOL in the core, then you'll be able to express a first-order theory. If you have FOL + set theory in the core, then you'll be able to express much more, because FOL + set theory is a poorly engineered replacement for simple type theory. If you attempt to have just a core FOL, and then build set theory on top of that, and then use that set theory as the foundation to develop the rest of math, things might get ugly, depending on what else is in the core. In cases where this works, there is good meta-level support in the core. $\endgroup$
– Andrej Bauer
Commented May 8 at 11:07
$\begingroup$ Regarding "FOL + categories", I suppose that would be some sort of Lawvere-style (such as discussed by McLarty here. Yeah, that could work, alhtough I am personally a fan of dependently typed formalisms, I think they fit more closely the reality of practice. $\endgroup$
– Andrej Bauer
Commented May 8 at 11:10

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