Talk:Axiom of Choice
Contents
Not really controversial anymore
Any proof which uses the axiom of choice can be transformed into a proof that doesn't. Granted, it will be a somewhat more complicated proof, but it always works, and that's a fact. That is the reason that AC is much less controversial these days than it was, in the early 1900s.
There is a complete explanation of the process and the proof that it's reliable here.
Also, the profoundly intuitive trichotomy is equivalent to AC, so be careful what you call controversial. BenjB 20:29, 27 January 2008 (EST)
Actually, the Axiom of Choice has been proven independent of ZF, so there is no such transformation of a proof. Otherwise, "prove" AC as follows:
1. Axiom of Choice | Reason: Axiom of Choice
Then transform it to not need AC. Result: AC proven in ZF,so ZFC=ZF. But AC proved independent of ZF. Therefore, no such transformation exists. QED SamSamson 21:50, 8 June 2008 (EDT)
This axiom actually can prove things which are (provably) unprovable without it; an example is Zorn's lemma. (Funnily enough, although Zorn is a guy's name, it's also a German word meaning "rage" ;-) --AMackenzie 14:29, 2 August 2008 (EDT)
- The Axiom of Choice can also be used to "prove" absurdities, as explained by the entry here.--Aschlafly 14:52, 2 August 2008 (EDT)
It is a wonderful thing in mathematics when an absurdity is proven - it expands the consciousness. Either the "absurdity" isn't as absurd as once thought (the Earth is a ball, not a plain; there are just as many whole numbers as there are fractions), or some axiom needs to be rethought, or you've made a mistake. Whatever, you can learn from it.--AMackenzie 17:45, 2 August 2008 (EDT)
- Banach-Tarski isn't absurd; it's just counterintuitive. -CSGuy 22:21, 2 August 2008 (EDT)
- It is absurd. Indeed, the point of the "proof" was to show how absurd it is.--Aschlafly 23:14, 2 August 2008 (EDT)
- It's absurd - no, it's not - yes, it is... As "being absurd" in this context is just a personal opinion, the phrase counter-intuitive seems to fit better. DiEb 10:28, 3 August 2008 (EDT)
Willard quote
I "prettied up" the quote from General Topology, but in context, Willard might not be the best person to quote after saying "many mathematicians reject the Axiom of Choice". Immediately before describing the axiom, he writes:
The following axiom is assumed by most mathematicians when they need it, to the unremitting disgust of a few.
And after the text quoted in the article:
The status of the axiom of choice bears some resemblance to that of the continuum hypothesis, with some differences. It, too, is known to be independant of the other axioms of set theory (that is, it or its negation can be consistently assumed), but it enjoys the status of an accepted part of the theory of sets in the minds of most modern mathematicians; that is, the intuition of almost all mathematicians now is that the axiom of choice should be assumed when needed without hestitation. Moreover, it is usually clearer that, where it is used, it is needed, so that its presence does not usually provoke the same frenzy of attempt to eliminate it.
Wandering 23:57, 2 August 2008 (EDT)
My edits
- I was unable to find references saying that the existence of a basis for every vector space and the existence of subsets of the real line without well-defined Lebesgue measure are equivalent to the Axiom of Choice. I did find references saying that the Axiom of Choice implies these, but that's not the same thing. -CSGuy 21:35, 26 November 2008 (EST)
The Axiom of Choice is not equivalent to the existence of non-Lebesgue measurable subsets of the real line and so I have removed this claim from the article. Intuitively one would not expect these to be equivalent. The existence of a non-Lebesgue measurable subset of the real line only says something about cardinals up to and including the continuum. However the Axiom of Choice makes a statement about all cardinals (in fact it is typically invoked when describing cardinals). AndyJM 14:27, 27 November 2008 (EST)
- I got curious and actually looked into this. The Axiom of Choice does imply the existence of non-Lebesgue measurable subsets of the real line. However, the existence of non-Lebesgue measurable subsets of the real line doesn't imply the Axiom of Choice. I found a paper called "The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set" and will quote the abstract for those who don't want to download a 500+kb document:
“ | Abstract. In this paper we show that the Axioms of Zermelo-Fraenkel set theory together with the Hahn-Banach theorem imply the existence of a non-Lebesgue measurable set. Our construction does not make any use of the Axiom of Choice. | ” |
- So the other direction of the claim is apparently wrong, and AndyJM was right to remove it. He was blocked for "removing information from articles" by the same user who also re-inserted the apparently wrong claim. This is exactly why I oppose this "Block first" approach: How can we improve this encyclopedia when we can get blocked and banned by whoever thinks we're wrong? I suggest an unblock. --AlanS 18:38, 27 November 2008 (EST)
- Alan, the Axiom of Choice yields many strange behaviors in mathematics. That's why it is still questioned and debated. If the statement is not equivalent to AC (which I will have to look deeper into to believe), then the correct behavior is not to remove the statement altogether but to change it to an implication. Edits that whitewash the Axiom of Choice's questionable role in mathematics are the sort of behavior you see at Wikipedia, not here (see Conservapedia:Critical Thinking in Math for a guide to our positions). I am reinstating the fact, but will tentatively change it to just a consequence of AC instead of an equivalence. -Foxtrot 18:50, 27 November 2008 (EST)
- I have no problems with stating that it's a consequence of AC, but that doesn't change (1) that you first re-inserted the apparently wrong information, (2) that you claimed on your talk page that "[i]t is well known that you need AC to create a non-Lebesgue measurable set of real numbers" and (3) that you blocked two users over this content dispute, one of them for five years. --AlanS 18:57, 27 November 2008 (EST)
- I reinserted valuable information that was lost by an editor intent to misrepresent the Axiom. If you persist in misrepresnting my actions in this matter, you may find yourself as the third Musketeer. -Foxtrot 19:01, 27 November 2008 (EST)
- I have taken this to the Abuse Helpdesk. --AlanS 19:24, 27 November 2008 (EST)
- I reinserted valuable information that was lost by an editor intent to misrepresent the Axiom. If you persist in misrepresnting my actions in this matter, you may find yourself as the third Musketeer. -Foxtrot 19:01, 27 November 2008 (EST)
- I have no problems with stating that it's a consequence of AC, but that doesn't change (1) that you first re-inserted the apparently wrong information, (2) that you claimed on your talk page that "[i]t is well known that you need AC to create a non-Lebesgue measurable set of real numbers" and (3) that you blocked two users over this content dispute, one of them for five years. --AlanS 18:57, 27 November 2008 (EST)
- Alan, the Axiom of Choice yields many strange behaviors in mathematics. That's why it is still questioned and debated. If the statement is not equivalent to AC (which I will have to look deeper into to believe), then the correct behavior is not to remove the statement altogether but to change it to an implication. Edits that whitewash the Axiom of Choice's questionable role in mathematics are the sort of behavior you see at Wikipedia, not here (see Conservapedia:Critical Thinking in Math for a guide to our positions). I am reinstating the fact, but will tentatively change it to just a consequence of AC instead of an equivalence. -Foxtrot 18:50, 27 November 2008 (EST)
- I'll look into this further but upon first glance agree with Foxtrot. Wikipedia, not here, is the place to pretend falsely that the Axiom of Choice cannot be questioned. It is questioned, it is disfavored, and it does sometimes yield absurd results. Further comments welcome here, and if anyone thinks someone was improperly blocked then let's look at all his edits here.--Aschlafly 19:13, 27 November 2008 (EST)
- Nobody claimed anything like that in this case. However, AndyJM removed a claim that was apparently wrong, and he was blocked for it. And BRichtigen was blocked for not simply ignoring this issue once Foxtrot lost interest. --AlanS 19:24, 27 November 2008 (EST)
- I'll look into this further but upon first glance agree with Foxtrot. Wikipedia, not here, is the place to pretend falsely that the Axiom of Choice cannot be questioned. It is questioned, it is disfavored, and it does sometimes yield absurd results. Further comments welcome here, and if anyone thinks someone was improperly blocked then let's look at all his edits here.--Aschlafly 19:13, 27 November 2008 (EST)
- AndyJM repeatedly deleted insights from entries, without explaining his deletions. That can be highly destructive if allowed to continue. We can debate the validity of his deletions, but at first glance AndyJM appears to me to have been wrong. Certainly his style was inappropriate in failing to explain or discuss his deletions. His block can be undone if you disagree, but more discussion would be needed about his specific edits.--Aschlafly 20:09, 27 November 2008 (EST)
- Thank you, Andy, for considering my reasoning. The reason for the block of AndyJM wasn't simply his edits to this entry, but his removal of multiple portions of several mathematics entries. I had looked at a few and saw that he was removing good material without explanation and so he was being destructive. -Foxtrot 20:24, 27 November 2008 (EST)
- Could you at least unblock BRichtigen? He already kicked off an analysis, but then you blocked him. I will paste the relevant exchange from your talk page:
- Thank you, Andy, for considering my reasoning. The reason for the block of AndyJM wasn't simply his edits to this entry, but his removal of multiple portions of several mathematics entries. I had looked at a few and saw that he was removing good material without explanation and so he was being destructive. -Foxtrot 20:24, 27 November 2008 (EST)
- Yes, of course, he was removing wrong information from articles
- AC is implies the existence of non-Lebesgue-mb sets on an intervall, but isn't equivalent to it.
- This, he outlined on the corresponding talk page
- Next removal: An irrational number is formally defined to be the limit of a Cauchy sequence of rational numbers. A real number can be formally defined to be the limit of a Cauchy sequence of rational numbers. For example, a constant sequence is Cauchy and will lead to a rational number...
- involvoling -> involving - no offense there, I presume
- The cardinality of such a set would be denoted by the Hebrew letter: He was right to remove this, too: without an index doesn't make much sense...
- The continuum is called so because it was the first (and most prominent) continuous set studied by mathematicians. What does this mean? Remove it, I'd say...
- All his entries were thoughtful and improve CP. --BRichtigen 16:41, 27 November 2008 (EST)
- I disagree with his removals and your recent edits to the Obama article as well as your past history do not instill confidence in what you are saying. Typos can be fixed, but it's wrong to remove information along with the typos. -Foxtrot 16:44, 27 November 2008 (EST)
- It's too late for me to do my own analysis right now, but I will see if I can add something tomorrow. In the meantime, you could do your part to show how AndyJM's edits were bad enough to warrant a five-year ban. We showed that at least one of his removals was entirely justified because it removed wrong information. I wouldn't be surprised if his other edits were okay, too. --AlanS 22:04, 27 November 2008 (EST)
- You have not shown how one of his removals was entirely justified. In fact, the removal was not justified. He removed valuable information from the article, rather than reworking it into a more acceptable form as it is now. He chose to remove entirely rather than to seek advice, and he did this repeatedly. His other edits removed valuable information as well, and if you and BRichtigen can't see that then I'm not sure either of you should be editing mathematics articles here. -Foxtrot 05:26, 28 November 2008 (EST)
- It's too late for me to do my own analysis right now, but I will see if I can add something tomorrow. In the meantime, you could do your part to show how AndyJM's edits were bad enough to warrant a five-year ban. We showed that at least one of his removals was entirely justified because it removed wrong information. I wouldn't be surprised if his other edits were okay, too. --AlanS 22:04, 27 November 2008 (EST)
- I disagree with his removals and your recent edits to the Obama article as well as your past history do not instill confidence in what you are saying. Typos can be fixed, but it's wrong to remove information along with the typos. -Foxtrot 16:44, 27 November 2008 (EST)
- And one for Andy...
- AndyJM repeatedly deleted insights from entries, without explaining his deletions.
- It has been pointed out that (for example) his edit to this article was correct - the claim had been wrong. So it hadn't been an insight, but an error.
- That can be highly destructive if allowed to continue.
- In my eyes, it's way more destructive to insert wrong information and to use block powers to intimidate and silence the opposition.
- We can debate the validity of his deletions, but at first glance AndyJM appears to me to have been wrong.
- At least in this case, he had been right. And BRichtigen tried to discuss the other edits, but Foxtrot wasn't in the mood and instead won the discussion by means of blocking.
- Certainly his style was inappropriate in failing to explain or discuss his deletions.
- He explained at least this deletion here, but Foxtrot removed the explanation for some reason instead of, you know, discussing about its validity. He also summed up his reasoning in the edit summaries at least.
- His block can be undone if you disagree, but more discussion would be needed about his specific edits.
- It would be immensely helpful if you could unblock BRichtigen. He already started such a discussion (I pasted a part of it here), and his knowledge might prove to be useful. I can (and will) take a closer look myself tomorrow, but having another experienced editor handy would lighten my load. --AlanS 22:13, 27 November 2008 (EST)
- And one for Andy...
- OK, done per your request. Thanks and let's see what happens next.--Aschlafly 22:31, 27 November 2008 (EST)
- It has been pointed out several times to you that his edit to the article was not correct, that he was not justified in removing the content. The content has remained in the entry and is valuable to it -- Andy agrees with this being another bizarre result from the Axiom of Choice and it benefits the article to LEAVE IT IN.
- Let me add that there's no need for your sarcasm and false personification of my "mood" and how I deal with editors. When an editor is removing valuable content from multiple entries, then they are blocked. When another editor refuses to follow a more senior editor's better judgment and persists on the point despite warning, they are blocked. You, my friend, are following the same path of haranguing behavior and that is why I have warned you that it may cause you to be blocked (and probably should have). But instead you're mischaracterizing this as "threats". I don't block often, but when I do it is with sound judgment. Enough time has already been wasted on this whole matter -- in the time I spent today dealing with your whining I could have made several worthwhile contributions and probably you could have too. -Foxtrot 05:26, 28 November 2008 (EST)
- It has been proven that his edits were correct. An implication is not equivalent to being equivalent. AlanS has taken quite an effort to improve the article and to explain his changes here on the talk page. His 90/10 block comes as a surprise to me. --BRichtigen
Latest edit
What's the point of this edit: to illustrate that another axiom also results in an absurdity? It strikes me as wrong to try to justify the AOC by pointing out that an absurdity can be reached without using it also:
- "Another seemingly absurd consequence of the Axiom of Choice is that there are subsets of the real line which do not have a well-defined Lebesgue measure. However, it is also possible to construct such a subset without using the Axiom of Choice.^{[1]}"
--Aschlafly 19:17, 27 November 2008 (EST)
- All other examples in that section seem to REQUIRE the AC. The subset thing doesn't. My edit doesn't justify the AC, but it points out that the nature of that absurdity isn't a direct consequence of the AC - the AC can just be used to construct it easily. --AlanS 19:28, 27 November 2008 (EST)
- But your distinction seems misplaced. The meaningful point is that AC proves an absurdity. It is irrelevant that another axiom may also produce the absurdity.--Aschlafly 19:46, 27 November 2008 (EST)
- Even if the AC was completely blown out of the water somehow, the subset issue would still exist. In a section where practically every other example depends on the AC, that is a distinction worth pointing out. But I rephrased it again. Does this please you more? I'm not here to defend or fight, but the omission of the proven possibility of constructing the subsets without AC had been notable, so I fixed it. --AlanS 21:56, 27 November 2008 (EST)
Deleted
Take it to Citizendium or Wikipedia. We don't need arcane stuff like this. First, write an article on how to prove a theorem in geometry or symbolic logic. --Ed Poor ^{Talk} 12:18, 4 December 2008 (EST)
- Arcane stuff like what? The article's gone. HelpJazz 12:47, 4 December 2008 (EST)
- Granted, the AC isn't trivial stuff, but it should have it place here, especially as quite a few sites link to it:
* Pierre de Fermat (← links) * Fermat's Last Theorem (← links) * Mathematics (← links) * Well-Ordering Theorem (← links) * Lebesgue measurable (← links) * Axiom of choice (redirect page) (← links) o Tychonoff theorem (← links) o Countable (← links) o Talk:Fermat's Last Theorem (← links) o User talk:Aschlafly/Archive34 (← links) o User:SamHB/Mathematical paradoxes (← links) * Real analysis (← links) * User talk:RSchlafly/Archive1 (← links) * Conservapedia:Critical Thinking in Math (← links) * Constructive proof (← links) * Banach-Tarski Paradox (← links) * Mathematical formula (← links) * Algebraic closure (← links) * User talk:Aschlafly/Archive34 (← links) * User talk:Foxtrot (← links) * Proof by induction (← links) * Axiom of empty set (← links) * Forcing (← links) * Talk:Examples of Bias in Wikipedia/Archive11 (← links) * Undecidable (← links) * Zermelo-Fraenkel (← links)
And A. Schlafly shows an interest in the AC - it is part of the proposed program of "critical thinking in math": This shows that it should be accessible for interested laymen, whom this course targets. The whole debate about elementary vs. non-elementary proofs cannot be understood without it. Please restore it. --BRichtigen 15:37, 4 December 2008 (EST)
- I've given you guys plenty of time to fall into step on this. Over and over I've been promised simple, easily understandable articles about basic math concepts. What do I get? Stuff that most freshman math majors can barely understand.
- I will wait three more weeks. If no one wants to help write the basic articles, I will clean house and start fresh: with How to count to 10. --Ed Poor ^{Talk} 19:24, 4 December 2008 (EST)
- (blush) Unless, of course, articles like this are too important to delete. :-( --Ed Poor ^{Talk} 19:26, 4 December 2008 (EST)
- Sorry about that, Ed. Thanks for your efforts.--Aschlafly 19:39, 4 December 2008 (EST)
Guises
Regarding the use of "guises" in the Instruction section, this is totally legit. When students begin using the Axiom of Choice, they are rarely told that it is the Axiom of Choice, and rarer still are they told the bizarre consequences and debate over the axiom. I myself was taught Zorn's Lemma first, and freely used it before I took some set theory and learned about the Axiom of Choice. Calling it a guise reflects the atmosphere of concealing the true nature of Zorn's Lemma in order to get students to use it and think it's not so bad. That wears down their suspicions for when they learn about the truly strange implications of the AC. It's not a bad analogy to compare it to drug dealers who get you hooked, and then you won't know how you could have done mathematics without it. To water this section down by simply stating that Zorn's Lemma is equivalent, is to ignore the pedagogical deception that is occuring (intentionally or unintentionally). -Foxtrot 05:06, 5 December 2008 (EST)
Note to Ed Poor about Pedagogy
Ed:
A couple of things before we begin:
- I come in peace. We have had bitter arguments in the past. I come in peace.
- The bulk of this message was written before the recent flap about deleting the axiom of choice article, though it does relate to that article. It was written just after your "grad students, stop showing off" comment. You need to understand this context. But it has suddenly become extremely timely. Please be patient.
- The article someone wrote giving the pronounciations of the numbers from 1 to 10 was (sorry about personal remarks!) utter garbage. That's not the way to go. I hope to explain. But I think things that you have written may have contributed to people thinking that's what you wanted. I hope to convince you that it isn't what you should want.
- I'm sure you are aware that the people to whom you issued the "three weeks" ultimatum are dead and gone.
There's a whole lot that I'd like to say on the subject of math pedagogy. (And that awful game of "wff 'n proof" :-) For background, you might want to look at some things I wrote a few days ago Talk:Boolean_algebra (you might not have seen it yet) and my draft of an article about paradoxes including some material incidentally relating to AC, in my sandbox User:SamHB/Mathematical_paradoxes.
I see that you are complaining about "grad students showing off" when they should be working on more fundamental issues of logic and proof theory. While I don't agree that they were showing off (they were simply trying to fix a disaster), I do agree that the page is a disaster. Among other things, it and its talk page have been block magnets from day one.
But we do need a page on that topic. You might want to look at my paradoxes draft to get an idea of the level that I consider appropriate. Note: It was not intended as a draft of AC. It just happens to talk about AC in another context.
Now, about proof theory. I already expressed in Talk:Boolean_algebra the view that mathematical logic should be considered a topic that is simply out of reach for CP. But, from what you have written above, you are interested in proof theory on a much more basic level. What you have written at Talk:Addition also shows that you want CP's math and science articles to reach all the way down to the fundamentals.
I'd like to argue that reaching too high and reaching too low are both pitfalls to be avoided. I've already discussed the issue of reaching too high, and that we have to be careful that, wherever we do reach, the path is solidly filled in. But, when we reach too low, we turn CP into a "textbook of common-sense notions". We are an encyclopedia. Something you look things up in when you already have basic knowledge of the subject, either to go on to the next step or to fill in gaps. We are not an introductory textbook. People do write such books (remember Dick and Jane?) and they are used in primary school. But I doubt that we have the expertise or time to write such things. (I speak only for myself. If you can do it, by all means do so, but I question the wisdom of following this path.)
Take the case of proof theory, which you raised both here and on the Boolean algebra talk page. The real "professional grade" topic is too high. (Except maybe for people who liked WFF_'N_PROOF :-) But you raised the question "how to prove a theorem of geometry". I submit that that's too low. The question in a student's mind that you would be trying to address would be something like "What does it mean for a proof of Pythagoras' theorem to be correct?" What does the proof mean? What does the statement mean? I submit that these are common-sense notions. 7th graders (or whatever) just need to be shown the statement and the proof, and they'll figure it out. Of course, they need common sense knowledge of how to interpret the sentences, but that isn't something we can teach them. As they go on in their mathematics education, they will become more adept at these issues. And, in graduate school, they may learn about formal proof theory. By the way, the existing Pythagorean_Theorem article is excellent. Among the best math pages we have. It needs no additional supporting articles from below.
SamHB 11:27, 5 December 2008 (EST)
- SamHB, what a surprise. Another "Open Letter to Ed Poor", albeit with a slightly different title. And as usual, it mixes in legitimate concerns with swipes at editors. I agree with you that it's important that the math articles don't get watered down to too low of a level. I'll totally give you that. But, I won't give you the swipe from aside about the Axiom of Choice article. It is not a poor article, in fact it's had numerous contributions to get it to its informative state, including several from Andy himself. We are one of the few places where students can learn about the implications of the Axiom of Choice before being compelled to use it in all their proofs. People who have tried to change that in the past, and we have kept steadfast to our position. If it's a block magnet, it's only because people like you don't agree with the article's position and want to change it to your own. -Foxtrot 14:29, 5 December 2008 (EST)
- Sam's open letter would be more interesting if I knew who he was. But I disagree with his high/low caution. We don't need a "one size fits all" approach to any topic. Anything that's hard to grasp, like genetics or molecular biology, will need an introductory article. Wikipedia has more than one article whose title is like, "Introduction to X". [1] Some topics can be covered in a single middle-level article. Usually, we have a simply intro - which tells the reader what we're going to tell him; then the we tell him (in the body). But a complex subject may require multiple articles. Wikipedia has over 100 articles trying to explain evolution. Which by the way shows it's not a simple topic. So we shouldn't dismiss the controversy over it by saying, "Well, the experts say it has been proven." We can prove that Galileo was right (and Aristotle was wrong) with a single, 10-second demonstration. So don't tell me science has to be complicated.
- Math is complex enough that it requires division into multiple levels. We need at least three, and I wouldn't be dismayed to end up with 4 or 5. Arithmetic starts with counting (and the definition of whole numbers vs. counting numbers (see Integer). Then we get addition and subtraction, which in turn provide the basis for multiplication and division. Next, we teach fractions and decimals. While we're doing this, we try to bring out rules such as the commutative and distributive laws, but kids aren't really tested on this.
- The next jump is to using variables ("unknowns"). Actually, we start by using blank spaces marked with underscores: 2 = 7 - ________ and then progress to clever shapes like a box! Somewhere between 6th grade and 9th grade we *gasp* take the astonishing step of using a letter like x for the unknown. It seems kids have to be somewhere around 12 years old to be able to do abstract reasoning like that (see Piaget). From there we make rapid progress through algebra (or "school algebra" if you like showing off), plane geometry, trigonometry and what high schools like to call "advanced mathematics" - meaning analytic geometry, differential calculus and integral calculus.
- Math majors at college and in grad school get to study on the next couple levels.
- Now what I am proposing - and I'll probably move this thread to the appropriate page - is that we focus on the first 2 or 3 levels before spending too much time on the upper couple of levels. --Ed Poor ^{Talk} 18:32, 6 December 2008 (EST)
Reversion explained
One does not have to "negate" the Axiom of Choice in order to reject its use in rigorous proofs.
- But the definition of AC is wrong. "For every nonempty set there is a function that chooses an element from each set." is false; for example, the set X = { { } } is a nonempty set that does not have a choice function. Sunda62 09:44, 12 April 2010 (EDT)