Timeline for What can be preserved in mathematics if all constructions are carried out in ZF?
Current License: CC BY-SA 4.0
55 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 18, 2022 at 1:18 | comment | added | Andreas Blass | The argument in my comment above is, I believe, in Platek, Richard A. Eliminating the continuum hypothesis. J. Symbolic Logic 34 (1969), 219–225. Despite the title, it also contains a lot about eliminating AC. | |
| Nov 18, 2022 at 1:13 | comment | added | Andreas Blass | @KamerynWilliams Shoenfield absoluteness is enough to show that $\Pi^1_4$ theorems of ZFC are provable in ZF. If ZFC proves $\forall X\exists Y\forall Z\exists W\phi$ (with $X,Y,Z,W$ ranging over reals and $\phi$ arithmetical) then to prove it in ZF, let an arbitrary $X$ be given, work in $L[X]$ (which satisfies ZFC) to get some $Y\in L[X]$ satisfying $\forall Z\exists W\phi$ in $L[X]$. By Shoenfield absoluteness, the same $Y$ works in the "real world". | |
| Sep 25, 2022 at 19:00 | answer | added | Sam Sanders | timeline score: 9 | |
| Sep 24, 2022 at 13:49 | answer | added | Martin Väth | timeline score: 11 | |
| Sep 23, 2022 at 12:33 | history | edited | Sergei Akbarov | CC BY-SA 4.0 |
deleted 82 characters in body
|
| Sep 22, 2022 at 21:51 | comment | added | Julia Williams | Minor comment, but no one's brought it up yet: Absoluteness results from set theory imply that AC cannot be necessary to prove sufficiently simple statements. For example, Shoenfield's absoluteness theorem implies that if you can prove, using AC, a statement of the form "for all $X$ there is a $Y$ such that $\varphi(X,Y)$" where $X$ and $Y$ range over sets of integers and $\varphi$ only quantifies over integers, then you can prove it without using AC. Many theorems about countable objects can be put in this form—my favorite example here is Hindman's theorem—so they're all preserved in only ZF. | |
| Sep 22, 2022 at 0:06 | comment | added | Sergei Akbarov | @AndrejBauer I corrected the question. I will not touch the comments, because there is no possibility to edit them, and besides this it seems to me they are not very important. | |
| Sep 22, 2022 at 0:04 | history | edited | Sergei Akbarov | CC BY-SA 4.0 |
I keep my promise to edit the text
|
| Sep 21, 2022 at 16:24 | comment | added | Z. M | A technical remark about a previous comment: compactness (without AC) seems to be defined in terms of open covers. Instead of nets, you can use filters, which leads to an equivalent definition without AC (nets are somehow choicy by the external nature, but filteres are intrinsic). | |
| Sep 21, 2022 at 15:33 | comment | added | Sergei Akbarov | @AndrejBauer ok, I'll correct the text, but later, because I was working today, and I am a bit tired. | |
| Sep 21, 2022 at 15:19 | comment | added | Andrej Bauer | @SergeiAkbarov: point taken. As for rewriting, just water down statements in which you imply, suggest, guess, or predict that horrible things happen once we remove the axiom of choice, because that's not true and it's tiresome to listen to such sentiments. | |
| Sep 21, 2022 at 14:50 | comment | added | Sergei Akbarov | @AndrejBauer I think our misunderstanding of each other comes from the fact that we hear discussions of these issues in different companies. You, as a specialist, hear the arguments of other specialists, and this does not cause rejection, because specialists understand each other. And I, as a non-specialist, hear this mostly from other non-specialists who follow the meaningfulness of what was said very poorly, and this revolts me. And some of that outrage is reflected in my posts in the MO. | |
| Sep 21, 2022 at 14:45 | comment | added | Sergei Akbarov | @AndrejBauer I can rewrite, but I actually don't feel what is wrong. You can suggest. | |
| Sep 21, 2022 at 14:43 | comment | added | Sergei Akbarov | @AndrejBauer "I guess" is a speculation, not a judgment. And "I foresee" was after Asaf's explanation about $\mathbb R$ without AC, this already gives enough grounds for judgment. Because it crosses out all the theorems from the standard list of questions for the analysis exam. | |
| Sep 21, 2022 at 14:41 | comment | added | Andrej Bauer | @SergeiAkbarov: so I think this would make a fine question if you edited it so that you genuinly ask the question without prejudice and insinuation as to what the conclusion of the answer must be. | |
| Sep 21, 2022 at 14:40 | answer | added | Andrej Bauer | timeline score: 8 | |
| Sep 21, 2022 at 14:30 | comment | added | Andrej Bauer | @SergeiAkbarov: and another one was "I guess that some elementary facts can be preserved in algebra, but, for example, in topology, I would be very surprised if something intelligible was preserved." You make your judgements before you even find out what the evidence is. | |
| Sep 21, 2022 at 14:29 | comment | added | Andrej Bauer | @SergeiAkbarov – Here is an off-putting remark of yours: "I foresee that when someone honestly writes a textbook on such a topology, people will stop arguing about the axiom of choice, being horrified by the picture that has opened up". The confirmation bias is amazing. | |
| Sep 21, 2022 at 14:02 | comment | added | Sergei Akbarov | To the people voting to close this question: wouldn't it be more honest to leave this thread for non-specialists, among whom certainly there are people who (like me) are interested in textbooks on analysis in ZF? | |
| Sep 21, 2022 at 13:39 | comment | added | Sergei Akbarov | You could notice that this is actually a big problem: to understand what people in a club mean when they say something, it often happens that you need to ask them a question that they by some strange regularity find indecent. And sometimes this leads to wrong quotations. | |
| Sep 21, 2022 at 13:39 | comment | added | Sergei Akbarov | @AndrejBauer I did not say this: "all sorts of mathematics "breaks down"", nor this: ""nothing can be done" without the axiom of choice". And I also have eyes and notions of decency, and it's not my fault that the simple-minded question "what happens if AC is removed from the list of axioms" is irritated by people constantly discussing "what happens if AC is removed from the proof of X, Y, Z". | |
| Sep 21, 2022 at 6:33 | comment | added | Andrej Bauer | @SergeiAkbarov: you confidently made a host of claims about how all sorts of mathematics "breaks down" and "nothing can be done" without the axiom of choice, which is simply false. I would urge a more cautious approach that doesn't irk people who actually know what can be done without AC and will simply ignore you as they're tired fighting this particular combination of ignorance and arrogance, driven by unchecked confirmation bias. | |
| Sep 21, 2022 at 5:38 | comment | added | Sergei Akbarov | @AsafKaragila as far as I understand, in this "constructive topology", the definition of compact space breaks down into two non-equivalent ones: the definition by open sets is not equivalent to the one where nets are used. That is, the same effect is manifested here as in constructive analysis by Kushner: definitions break down into different non-equivalent ones. I foresee that when someone honestly writes a textbook on such a topology, people will stop arguing about the axiom of choice, being horrified by the picture that has opened up. | |
| Sep 20, 2022 at 19:42 | history | became hot network question | |||
| Sep 20, 2022 at 19:31 | comment | added | Sergei Akbarov | Ah, the idea is that the theory is changed, but the examples of topological spaces constructed from real numbers do not disappear, they are preserved but with some different properties. | |
| Sep 20, 2022 at 18:14 | comment | added | Sergei Akbarov | @AsafKaragila so if I understand correctly, this means that the theory of real numbers is not changed when we throw away the axiom of choice? | |
| Sep 20, 2022 at 18:11 | comment | added | Asaf Karagila♦ | @SergeiAkbarov: It is very not true. No. It might be wise, before starting with mathematics without AC, to learn a bit of wet theory without AC. For that Jech's Axiom of Choice book is pretty decent. Azriel Levy is known to keep track of it, so I imagine that his book in Basic Set Theory would be good as well. | |
| Sep 20, 2022 at 18:00 | answer | added | Timothy Chow | timeline score: 22 | |
| Sep 20, 2022 at 17:50 | comment | added | Sergei Akbarov | @AsafKaragila (1) I did not tell that real numbers manifest themselves somehow in the definition of topological spaces; (2) but I used to think that in the standard construction of real numbers in axiomatic set theories like ZFC (in the sequence $\mathbb{N}\to \mathbb{Z}\to \mathbb{Q}\to \mathbb{R}$) the axiom of choice is used, for example, when people use the fact that the set of finite ordinal numbers $\mathbb{N}$ is well-ordered. Is that not true? | |
| Sep 20, 2022 at 17:35 | review | Close votes | |||
| Sep 30, 2022 at 3:03 | |||||
| Sep 20, 2022 at 17:35 | comment | added | Asaf Karagila♦ | (1) Where in the definition of a topology do the real numbers come in? (2) Where in the definition or construction of the real numbers does the axiom of choice come in? | |
| Sep 20, 2022 at 17:24 | comment | added | Sergei Akbarov | @RyanBudney I expected that people who are arguing about "constructive proofs" have already accumulated a list of examples. I didn't see it. | |
| Sep 20, 2022 at 17:19 | comment | added | Sergei Akbarov | @AsafKaragila, I don't see problems with the definition of topological spaces. But if we throw away the theory of real numbers (and the axiom of choice) what examples of topological spaces remain? | |
| Sep 20, 2022 at 17:18 | comment | added | Ryan Budney | This sounds a little bit like a "tell me about logic" question. It might make more sense to single out a specific theorem and ask what remains of it, when you use other, specific logical foundations. | |
| Sep 20, 2022 at 16:50 | comment | added | Asaf Karagila♦ | Where do you think topology is being used when you define topological spaces? | |
| Sep 20, 2022 at 16:02 | comment | added | Sergei Akbarov | @AsafKaragila in my understanding, on the contrary, I try to remove philosophy from mathematics. All mathematical theories are built inductively: first people list the rules of the game, and then they play the game. Only in logic everything is not like in the other mathematics. But still, what examples of topological spaces remain if we exclude the axiom of choice? | |
| Sep 20, 2022 at 15:49 | comment | added | Asaf Karagila♦ | @SergeiAkbarov: I think you are veering too far into philosophy, even for my taste. And I'm not even sure that this is still mathematical philosophy at all, rather than just general philosophical approach to the logic used in proofs and formal statements. I don't know if this is a fruitful area of mathematical work, to be honest. I'm happy to have discussions about philosophy, as most people who had a few beers with me will testify, but to say that somehow the fact we omitted an axiom that was irrelevant to the definition requires us to re-examine everything? That might be going too far. | |
| Sep 20, 2022 at 15:27 | history | edited | Sergei Akbarov | CC BY-SA 4.0 |
deleted 20 characters in body
|
| Sep 20, 2022 at 15:11 | comment | added | Sergei Akbarov | Also, when you find a "constructive proof" of X, the viewer can't check that the Gentzen tree of your reasoning doesn't really contain the axiom of choice, because this tree breaks at the lemmas you use. In a "linearly built" theory, the viewer can look at the proof of the previous results and make sure that everything is clean there, but in your situation you don’t have a "linear theory". | |
| Sep 20, 2022 at 15:11 | comment | added | Sergei Akbarov | What is surprising in these debates is the willingness of people to discuss mathematical questions beyond the framework of the mathematical approach, metaphysically, in Hegel's style. One would expect that a person first constructs a theory with examples and problem statements that would convince the audience that this is really interesting, and only then he proves deep theorems about the objects of this theory. But here we see a situation where even the list of examples is not clear what. | |
| Sep 20, 2022 at 14:45 | comment | added | Sergei Akbarov | @AsafKaragila thank you for the references, I'll look at this. As to compact subsets I even don't see if this question still makes sense in the situation when we drop the axiom of choice. After all, in this case, the (usual) theory of the real numbers must also be thrown out, and I do not even understand what kind of examples of topological spaces remain. Does it make sense to be interested in their properties, if the examples known to you disappear, and you don’t understand what others are? If these examples are anything trivial or absurd, how can such a theory be interesting? | |
| Sep 20, 2022 at 14:16 | comment | added | Asaf Karagila♦ | @SergeiAkbarov: Certainly you can find a lot of information as to what can happen in algebra, as I remarked, Herrlich's book contains a lot of information. On all three fields, actually. I've also mentioned Schechter's book in a previous comment, as a source for analysis, etc. As for the second comment, I'm not sure what is unclear. Asking if we can prove that compact subsets of the real line are closed can be done with or without the axiom of choice, but do you really need to rewrite classical analysis and basic topology in ZF for that? | |
| Sep 20, 2022 at 13:43 | comment | added | Sergei Akbarov | @AsafKaragila and as a specialist in this field you could clarify this: "When a person asks whether the statement X is true without the axiom of choice, does he mean to remove the axiom of choice from the whole theory in which he formulates this statement, or only from the proof of X?" | |
| Sep 20, 2022 at 13:42 | comment | added | Sergei Akbarov | @AsafKaragila: I did not expect that somebody rewrites "all of mathematics" without choice. This could be done in some fields. As an example, Kushner shows what the intuitionists point of view gives in analysis. I would think, that the people who discuss the "validity of the application of the axiom of choice here and there" could do something similar in this direction. Does your "answer is obviously not" mean that there are no reviews of what happens in Algebra, Topology, or Analysis? | |
| Sep 20, 2022 at 13:39 | history | edited | Sergei Akbarov | CC BY-SA 4.0 |
added 86 characters in body
|
| Sep 20, 2022 at 13:08 | comment | added | Sergei Akbarov | @A.S. linear algebra uses the results of the theory of real numbers which is usually constructed inside the axiomatic set theories like ZFC. I think it's impossible to say that linear algebra does not use the axiom of choice. | |
| Sep 20, 2022 at 13:01 | comment | added | Asaf Karagila♦ | @A.S. And even anafunctors will fail, eventually, in ZF, as you can see in mathoverflow.net/q/264585/7206, and even if they don't, the question still feels broad. | |
| Sep 20, 2022 at 12:54 | comment | added | user164898 | @AsafKaragila: About "Did anyone ever redo all of mathematics without choice?": I have the impression that "anafunctors" in category theory are motivated by a desire to do something along those lines, at least for the many arguments in mathematics which have a clear category-theoretic formulation. My impression (as an outsider) is that "anafunctors" are a variant of functors, which behave well in the absence of AC, and replacing functors with anafunctors in many familiar category-theoretic arguments lets classical arguments go through without AC. ncatlab.org/nlab/show/anafunctor | |
| Sep 20, 2022 at 12:48 | comment | added | user164898 | I like the question. A typical first course in undergraduate linear algebra doesn't use AC at all, as far as I know, since all the vector spaces considered there are finite-dimensional. In a typical undergraduate course in abstract algebra, I think you first encounter AC when you show that every commutative ring has a maximal ideal. I wonder how much of the standard material covered in such a course survives without AC. Presumably you have a lot of similar statements but with extra hypotheses, e.g. "Let $R$ be a commutative ring which has a maximal ideal." | |
| Sep 20, 2022 at 12:35 | comment | added | Asaf Karagila♦ | Yes, but you're asking "Hey, did anyone ever redo all of mathematics without choice?" and the answer is obviously not. There are some books, like Schecther's Handbook of Analysis and its Foundations which discuss some aspects of, well, analysis; Fremlin's Measure Theory has a part at the end of vol. 5 dedicated to, well, measure theory; and surely similar books exist in other fields. Herrlich has The Axiom of Choice which contains many possible disasters that may occur without choice throughout mathematics. But none of those is even remotely complete in any sense of the word. | |
| Sep 20, 2022 at 12:12 | comment | added | Sergei Akbarov | @AsafKaragila I have added the tag "reference-request". | |
| Sep 20, 2022 at 11:58 | history | edited | Sergei Akbarov | CC BY-SA 4.0 |
added 2 characters in body
|
| Sep 20, 2022 at 11:52 | history | edited | Sergei Akbarov |
edited tags
|
|
| Sep 20, 2022 at 11:49 | comment | added | Asaf Karagila♦ | This is way too open ended. | |
| Sep 20, 2022 at 11:40 | history | asked | Sergei Akbarov | CC BY-SA 4.0 |