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Stream: deprecated: logic

Topic: Axiom of Multiple Choice

Brian Pinsky (Jan 02 2021 at 21:49):

The nlab page for axiom of multiple choice (https://ncatlab.org/nlab/show/axiom+of+multiple+choice) refers to something like "the category of sets has enough projectives" (I'm not sure I fully understand it so it's best to see the definition there)

However, Jech's "the axiom of choice" book and some other set theory sources give what seems to be a different definition. I believe this definition is unrelated and would like to edit the nlab page to include it, but I wanted to make sure it wasn't equivalent to the nlab definition in a way I don't understand first.

The set-theoretic axiom I've seen is:
if $X$ is a set of non-empty sets then there is a function $f$ defined on $X$ such that for all $y\in X$, $f(y)$ is a finite proper subset of $y$

If this somehow follows from the definition on nlab, I'd appreciate if someone could explain to me how.
(For an example of this axiom in a set theory paper, see http://matwbn.icm.edu.pl/ksiazki/fm/fm50/fm50140.pdf, where it is also called $Z(\infty )$)

John Baez (Jan 02 2021 at 22:15):

Maybe someone here can help you, but I also urge you to repost your question to the nForum discussion on axioms of choice - that way @Mike Shulman, who wrote the nLab article on the axiom of multiple choice, will be sure to see your question.

John Baez (Jan 02 2021 at 22:17):

The nLab page for the axiom of multiple choice lists 3 formulations of this axiom, the first 2 being equivalent and the third being weaker. None of these says anything like "the category of sets has enough projectives".

John Baez (Jan 02 2021 at 22:19):

One big obvious difference between those formulations and what you're saying is that those formulations don't mention anything about finiteness.

Mike Shulman (Jan 02 2021 at 22:33):

Yes, I think that's a different axiom.

Mike Shulman (Jan 02 2021 at 22:34):

(I've actually been kind of remiss recently in keeping up with the nForum, but Zulip emails me when I get mentioned...)

Brian Pinsky (Jan 03 2021 at 19:06):

John Baez said:

Maybe someone here can help you, but I also urge you to repost your question to the nForum discussion on axioms of choice - that way Mike Shulman, who wrote the nLab article on the axiom of multiple choice, will be sure to see your question.

Okay; I haven't written on nForum before but I can post there too (although it seems mike has already replied here so I can safely assume these axioms are different and edit the page accordingly). I'm not aware of any nice categorification of the set theoretic axiom, which would feel fitting for an nlab page; maybe someone there would know.

Brian Pinsky (Jan 03 2021 at 19:17):

John Baez said:

The nLab page for the axiom of multiple choice lists 3 formulations of this axiom, the first 2 being equivalent and the third being weaker. None of these says anything like "the category of sets has enough projectives".

I guess it seemed related because the notion of a collection family seems like a weaker version of a projective object (it's like a projective family of objects instead; any one object collection family is a projective object). It's possible that intuition is wrong or unhelpful; I haven't really used the definition much.

Brian Pinsky (Jan 03 2021 at 20:12):

@Mike Shulman I've posted on nforum now, if you wanted to have more discussion there

Brian Pinsky (Jan 03 2021 at 21:11):

Update: The axioms are decidedly not equivalent. Jech's book construct's a permutation model where multiple choice fails, but all permutation models satisfy SVC (https://ncatlab.org/nlab/show/small+violations+of+choice), which implies the categorical axiom