Talk:Axiom of choice

Images at top
I like the new one, but it has the same technical problem as the old one. Take the colored item. No choice axiom needed for that. -- 10:11, 5 April 2017‎ (UTC)

X as a collection or set?
I'm still learning but shouldn't the boxed statement of the axiom refer to X as a collection, not a set?


 * In ZFC set theory, every entity is a set. Every collection is a set of sets. See pure set. JRSpriggs (talk) 21:02, 25 July 2021 (UTC)

Essay on reasons to use choice
I reverted this paragraph:
 * One may notice that many of the above statements (and also some of the statements in the next sections using weaker forms of choice) come in one of two following types: First, there are those statements which assert that a given property pertaining to a type of mathematical structure is closed under arbitrary Cartesian products (e.g. the Cartesian product of nonempty sets is nonempty, or the Cartesian product of connected topological spaces is connected). Second, there are those statements that assert the existence of a maximal element of some special collection of subsets in a given mathematical structure, and moreover assert that any such subset can be extended to a maximal one (e.g., the Hausdorff Maximal Principle, Krull's Theorem, and the assertion that every vector space has a basis). This pattern is not a coincidence, as can be seen by the following reasoning: To prove any of these statements when the Cartesian product (in the case of the first type) or the special collection of subsets (in the case of the second type) is finite, one simply can use finite induction on the size of these collections or indexing sets. However, to extend these statements to infinite cardinalities, one needs to go beyond finite induction to transfinite induction, and as mentioned earlier in the article, this in general requires the Axiom of Choice.

So the first objection is that, helpful or not, this seems to be a collection of personal reflections, not sourced to the literature.

Beyond that, there's a confusion here about transfinite induction. Transfinite induction per se does not rely on the axiom of choice. Not in any way whatsoever.

What does rely on the axiom of choice is inductive arguments that involving "picking" a particular element at each induction step. But the choice is used for the picking, not for the induction. We really have to be clear on this, because this is a common misconception. I haven't scoured the article to see if there's anywhere else this needs to be clarified. --Trovatore (talk) 19:31, 19 April 2022 (UTC)


 * Just to mention, I have seen the term "principle of transfinite induction" used in the literature to actually refer to forms of the Axiom of Choice such as Zorn's Lemma. For example, there is Terence Tao's classic text Analysis I. In Section 8.4 in this book, he uses the phrase "principle of transfinite induction" to refer to Zorn's Lemma. Also, the classic abstract algebra textbook by Dummit and Foote concedes in Appendix A that Zorn's Lemma can viewed as a sort of "infinite induction", i.e. a generalization of finite induction to infinite partially ordered sets. The point is that the principle of finite induction can be reframed in the context of partial orders, to essentially assert the existence of a maximal chain (or element) in any finite partially ordered set; Zorn's Lemma naturally extends this fact to infinite partially ordered sets. One must recognize that this is a different usage of the term "principle of transfinite induction" than is typically used in the context of ordinals. However, there are some merits in the way this term is used as above. Often, novices have a hard time understanding the intuition behind Zorn's Lemma, but hopefully presenting it as a sort of "infinite induction" can help aid understanding. Logic314 (talk) 22:22, 23 April 2022 (UTC)

Controversy
This part is misleading: “Although originally controversial, the axiom of choice is now used without reservation by most mathematicians,[3] and it is included in the standard form of axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC).”

That’s not what the reference says. The reference acknowledges that AC is the most contested axiom, and that it is less controversial in recent years. However it still remains very controversial, with many mathematicians avoiding its use. The other sentence is also misleading because AC is not automatically included in standard set theory. It is independent. That’s why there is ZF, ZFC, and ZF with negated C. 184.169.45.4 (talk) 19:24, 8 October 2022 (UTC)
 * Your fourth sentence in the second paragraph is not correct. AC is part of default set theory. --Trovatore (talk) 19:27, 8 October 2022 (UTC)
 * The text in the Wiki article is still not reflecting the citation, because it makes the claim that AC is not controversial anymore, when in fact it remains controversial. The cited source just says it is less controversial.  The text should be edited to remove bias and acknowledge the ongoing controversy. 107.194.133.235 (talk) 02:54, 11 October 2022 (UTC)

Citation for "all known models"?
The section "Possibly equivalent implications of AC" says "In every known model of ZF where choice fails, these statements fail too, but it is unknown if they can hold without choice." Can we include a reference (or set of references) for the claim that these statements fail in all known models of ZF where choice fails? 2601:547:C80:8010:E818:6850:DF4:9128 (talk) 18:24, 17 May 2023 (UTC)