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Mathematicians usually don't write "by the axiom of choice" when they prove some propositions even if they use the axiom of choice.

But they always emphasize the fact they use the axiom of choice when they prove Zorn's Lemma.

Why?

tchappy ha
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    I upvoted your question, but also voted to close it as a duplicate of the question that was linked in the fake comment above that I didn't write. I think that question will answer yours because Asaf Karagila's answer has a proof that Zorn's Lemma and AC are equivalent and Martin Sleziak's answer explains what it means for a theorem to be equivalent to something that's normally taken as an axiom. – Greg Nisbet Jan 11 '23 at 05:02
  • @GregNisbet Thank you very much for your link. I am reading "Introduction to Set Theory and Topology" (in Japanese) by Kazuo Matsuzaka. In this book, the author wrote Zorn's Lemma is equivalent to the axiom of choice. I just wonder why mathematicians emphasize the fact they use the axiom of choice only when they prove Zorn's Lemma. – tchappy ha Jan 11 '23 at 05:04
  • I am glad if mathematicians always emphasize the fact they use the axiom of choice when they use the axiom of choice. – tchappy ha Jan 11 '23 at 05:07
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    Martin Sleziak's answer answers that exact question. When you prove something is equivalent to the axiom of choice, you are temporarily working in a weaker system such as ZF. If you were not working in a weaker system than ZFC, then the axiom of choice would just be true and thus equivalent to any true statement. The author is emphasizing the axiom of choice because they are not assuming it in the background. – Greg Nisbet Jan 11 '23 at 05:08
  • @GregNisbet Thank you very much for the comment. – tchappy ha Jan 11 '23 at 05:09
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    Honestly, I've seen books call out when they use AoC (or Zorn's lemma), usually with a sense of: What follows is an incredibly general theorem. We could prove a version sufficient for all the actual examples we're going to use without assuming it, but we'll present the most general version. – JonathanZ Jan 11 '23 at 05:15
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    This was closed as a duplicate of a question that has absolutely nothing to do with what is being asked here. Here the question is why people don't bother saying that they use AoC except when proving Zorn's Lemma and it is being closed as a duplicate of a question which asks how one can show that AoC and Z'sL are equivalent. There is not even a similarity between the two questions... Great. – Mariano Suárez-Álvarez Jan 11 '23 at 05:38
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    Not only this is very clear from the question itself but was repeated by the poster in a comment. – Mariano Suárez-Álvarez Jan 11 '23 at 05:41
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    @MarianoSuárez-Álvarez Sorry. I'll vote to reopen. I honestly thought that's what the thrust of the question was: why are these two things equivalent. – Greg Nisbet Jan 11 '23 at 05:42
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    (and Martin's perfect explanation about using weaker systems and so on has nothing to do, because most people who prove things that depend on the axiom of choice would not be able to explain what a weaker system is...) – Mariano Suárez-Álvarez Jan 11 '23 at 05:49

2 Answers2

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First of all, many mathematicians will mention the Axiom of Choice when they use it. But then again, many others will not mention Power Set, Extensionality, Replacement, Separation, Union, or Infinity when they rely on those.

The reason for Zorn's Lemma to be more explicit is that we need to provide some "nontrivial structure" for it in the form of a partial order with certain properties, whereas the Axiom of Choice merely requires us to have a bunch of non-empty sets and in most cases it is obvious that the sets are not empty.

Not to mention that the vast majority of mathematicians are not interested in whether or not they used the Axiom of Choice, or if it was truly necessary. And, again, the same can be said about Power Set, Extensionality, etc. When the foundation of mathematics gets in your way of doing mathematics, it might be a good idea to ask yourself if you are doing mathematics or just studying its foundations.

Asaf Karagila
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  • Asaf Karagila, Thank you very much for your answer. – tchappy ha Jan 11 '23 at 10:58
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    Kinda surprised to see you write "doing mathematics or just studying its foundations". It's sometimes hard to be sure of irony in internet forum writing, but I assume that phrase comes with a self-deprecating smile? – JonathanZ Jan 11 '23 at 18:11
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    @JonathanZsupportsMonicaC: No, not really in an ironic tone, even though I do study the foundations of mathematics. The remark is more to the point that if you insist too much to refer to the foundations of mathematics, rather than doing mathematics, then perhaps you are interested in the research on the foundations of mathematics as much, or even more than, than the research on actual mathematics. – Asaf Karagila Jan 11 '23 at 23:16
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I'm going to challenge the assertion of the question. I've done mostly topology and analysis, and any time I've seen texts or lecturers (at the graduate level, say) prove something using the AoC (or the equivalent Zorn's Lemma, or the Hausdorff Maximal Principle), they've made a big deal of the fact.

I think it's because you have to haul in some non-standard machinery (like filters/ultrafilters, or posets/chains) to use these, and so they are mentioned to justify the construction (as AsafK mentioned). E.g. "We are going to use Hausdorff maximal principle, so to do this we define the following order ...".

And even if someone is using the AoC version, they will say "Okay, now we take a choice function for our collection ....", and the use of the phrase "choice function" signals to everyone that they are using the AoC. It's like Whitney Houston doesn't have to explicitly state "This is the part of the song where I will hold a high note for a long time" for all her listeners to know "this is where she holds a high note for a long time".

JonathanZ
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