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• CommentRowNumber1.
• CommentAuthorBrian Pinsky
• CommentTimeJan 3rd 2021

These two related principals are not the only axioms with this name. A more set-theoretic “axiom of multiple choice” also exists (which, as far as I can tell, is unrelated to these axioms). I added the set theoretic axiom and gave some references where it is used this way.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJan 3rd 2021

Again, I think it’s worth considering creating a separate page for this other notion you’re referring to. See my earlier comment in the other thread.

1. removing query box from article

+–{: .query} Mike Shulman: Are the first two the same? If not, why are they given the same name?

Peter LeFanu Lumsdaine: Yes, they are equivalent. For any $X$, given a collection family $\{D_c\}_{c \in C}$ including $X$, then the family $\{D_c\}_{(c \in C, f : D_c \twoheadrightarrow X)}$” is an inhabited collection family equipped with surjections to $X$. Conversely, given an inhabited collection family equipped with surjections to $X$, throwing $X$ into the family gives a collection family including $X$. =–

Anonymous

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