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This question is related to the answer of "Category theory from the first order logic point of view". They says:

Any model of this theory is a category. Unfortunately, however, that would beg the question of what to use as a model-construction language. First-order logic is typically associated with models constructed in a set theory such as ZF. That would defeat the whole purpose of this as an alternative foundation, as well as limiting us to "small" or "internal" categories.

I don't know fully what this mean. As described in this answer, axioms of category theory is formulated by first order logic and so it will seems that we can even treat models of axioms of category theory. One reason I think is ZF contradict the existence of a model such as $Set$(i.e the category of sets). Therefore I think if we assume ZFC+the existence of Grothendieck Universe then we don't need "small" or "internal" categories.Is it correct?

beginner_nt
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  • If the objects of a category form a set then there is no category of all sets in ZF. Adding axioms doesn't change this. – Zhen Lin Nov 03 '20 at 04:47
  • We think "a set" is "an element of the grothendieck universe" in ZFC+U(for short). In this case, there is category of all sets. – beginner_nt Nov 03 '20 at 05:10
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    There is no "the" Grothendieck universe. Every set is a member of some universe. – Zhen Lin Nov 03 '20 at 05:22
  • That is correct. It seems I have implicitly limit a situation, sorry for that. Ordinary, when we work in category theory at ZFC+U, we enlarge a grothendieck universe as much as we need. So if we aim the proof of a theorem then we can fix the grothendieck universe. In this case, do we need limiting to "small" or "internal" categories? – beginner_nt Nov 03 '20 at 06:09
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    Universes do not solve the small / large distinction. See here. – Zhen Lin Nov 03 '20 at 06:52
  • Thank you for answers. I will try to understand it. – beginner_nt Nov 04 '20 at 07:12

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