Talk:Axiom of choice

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 Field: Foundations, logic, and set theory

Well-ordering theorem or principle?[edit]

In the Quotes section, the first quote (by Jerry Bona) refers to the Well-ordering principle which is linked to the Well-ordering theorem, where we are warned that it is Not to be confused with Well-ordering principle. Which is it? If the quote is accurate but the reference should be the theorem (my suspicion), then this should be addressed explicitly. askewchan (talk) 14:42, 12 August 2014 (UTC)

The warning would not be there were there not a lot of people who have said "well-ordering principle" when they meant "well-ordering theorem" (or the reverse). Obviously, Jerry Bona is one of them. An explanation of that would be over-kill and distracting. JRSpriggs (talk) 17:46, 12 August 2014 (UTC)

Proof of Axiom of Choice[edit]

The Axiom of Choice assumes S_i are non-empty sets. If a set is given by enumerating its elements, we know when it is not empty and then we can select or choose one of its elements, say the first.

If a set is given by a property its elements share, then you cannot simply state the set is non-empty; you always have to prove it, meaning to identify at least one of its elements. For instance, S = \{x\in R|x>n,n\in N,\forall y\in R,y>n,x\leq y\} = \{x\in R|n\in N,x is the least element of (n,n+1)\} is the empty set as it has no elements.

The Axiom of Choice assumes S_i are non-empty sets, so we have already proven they are non-empty, meaning we have already identified at least one element of each S_i.

Then the choice function in the Axiom of Choice could be:

"From each non-empty set choose the first identified element."

Aurelian Radoaca (talk) 15:07, 31 August 2014 (UTC)

"Nonemty" is a premise. You don't prove those. You assume them. YohanN7 (talk) 16:46, 31 August 2014 (UTC)

The Axiom of Choice says S_i are non-empty sets, so you know that.

Given a set S, in order to know it is non-empty, you have to identify at least one element (or otherwise prove it is non-empty, in case you cannot identify an element), or else the set may be empty. Is it possible to know a set is non-empty and not being able to identify one element? In either case, in order to know a set is non-empty, you have to prove it.

The Axiom of Choice starts with the assumption that S_i are non-empty sets, so you know they are non-empty.

Aurelian Radoaca (talk) 17:45, 31 August 2014 (UTC)

Let S be the power set of the reals except for the empty set. Now you know they are nonempty. What is your choice function?
You should probably take this to Wikipedia:Reference desk/Mathematics. This page is for discussing the article. YohanN7 (talk) 18:04, 31 August 2014 (UTC)

Any union of countably many countable sets is itself countable[edit]

My undergraduate maths degree was most of three decades ago, so this must be tentative. But does “Any union of countably many countable sets is itself countable” really need anything more than plain ZF? It seems ‘obvious’ that this can be shown in much the same way that ℚ can be shown to be countable.

Of course I must be wrong. In which case please could this line acquire a source? (Or a sketch explanation under this talk comment.) Thank you. JDAWiseman (talk) 19:36, 29 November 2014 (UTC)

Further thought: what if each set is countable in uncountably many different ways, between which one can’t distinguish? Ahh. But maybe a few words or a source would help those confused as I was. JDAWiseman (talk) 19:46, 29 November 2014 (UTC)

This issue has been discussed before, repeatedly I think. For example, see Talk:Axiom of choice/Archive 2#countable sets. The upshot is that at least the axiom of countable choice is needed to conclude that a countable union of countable sets is countable. JRSpriggs (talk) 13:07, 30 November 2014 (UTC)
Excellent, thank you. I was wrong, I was correct, and you have proved me even more correct. My ‘like ℚ’ argument was wrong. My ‘how choose one of many orderings’ was correct. So far, I hope, we agree. But your archive link adds weight to my suggestion that “a few words or a source would help those confused as I was”. If multiple readers are confused by something in a Wikipedia entry, the entry could be improved.
What words? Perhaps add “(because it is necessary to choose a particular ordering for each of the countably many sets)”. I’m not insisting on those words, but please hear confused readers needing help — and it won’t be just those who have posted in the talk pages, JDAWiseman (talk) 15:49, 30 November 2014 (UTC)
That is a good wording. Feel free to add it to the article. JRSpriggs (talk) 08:17, 1 December 2014 (UTC)
Done. JDAWiseman (talk) 08:26, 1 December 2014 (UTC)


Is the statement "For any two nonempty sets X and Y, there is a surjection X->Y or a surjection Y->X." equivalent to the axiom of choice? GeoffreyT2000 (talk) 04:50, 26 February 2015 (UTC)

I think that the answer is "yes". It is similar to but more complicated than the proof for injections.
Clearly, the axiom of choice implies that one of the surjections exists because X would be equinumerous with an ordinal and Y would be equinumerous with another ordinal. The larger ordinal could be mapped onto the smaller one, so there would be a surjection between X and Y (one way or the other).
To go the other way, suppose X is an arbitrary nonempty set, we will show that it can be well-ordered which implies the axiom of choice.
Let Y=P(X)+, that is the Hartogs number of the powerset of X. Then there is no injection from the ordinal Y to the powerset of X. Thus there is no surjection from X to Y.
So by your hypothesis, there must be a surjection from Y to X. For each element x of X, find the least element y of Y which maps to x. Well order X according to the ordering on the y associated with each x in X. JRSpriggs (talk) 08:01, 26 February 2015 (UTC)

Quantum and Cosmological Axiom application[edit]

This article does not mention at all the quantum and cosmological contributions of the "axiom of choice". Also the quantum cryptography applications of it. -- (talk · contribs) 18:10, 14 April 2015‎ (UTC)

Are there any such applications? It seems unlikely to me. If you know of any, please provide a reference to a reliable source. JRSpriggs (talk) 19:57, 15 April 2015 (UTC)
Unlikely, yes! But there are results linking set theory and physics. Google for "Some Set Theories are More Equal" (I was unable to place a link here for some reason). The reference, by Menachem Magidor, is not published, but still probably reliable. There are even results linking the continuum hypothesis and Bell's theorem (see section 5 in ref). YohanN7 (talk) 11:42, 28 April 2015 (UTC)
We seem to be dealing with "philosophy of set theory" articles here; the question of whether those are "reliable", even if in otherwise reliable journals, is still open. (BTW, as a set theory expert, I assert that the connection between the continuum hypothesis and Bell's theorem is flawed, as the maps do no good unless measurable, and (here, as I'm not a physics expert, I cannot be sure), seem to have no physical significance unless they meet some continuity requirement.) — Arthur Rubin (talk) 17:04, 28 April 2015 (UTC)
I think Magidor's point is this (from his paper):
As to be expected we do not have any definite case in which different set theories have an impact on physical theories but we believe that the possibility that it may happen in the future is not as outrageous as it may sound.
Not outrageous, that is, just very very improbable. There may be flaws in Magidor's reasoning of course. As far as continuity requirement go, we don't even know for sure that space and time is "a continuous background", so it may be tricky to even define "continuity", i.e. a suitable topology on spacetime in which "continuity" make physical' sense.
I don't believe much of this, but it is intriguing that a notable set theorist has taken up the issue. And it is fun to speculate a little, even though this probably is the wrong forum for it. YohanN7 (talk) 17:50, 28 April 2015 (UTC)
The link to your suggested reference is here. JRSpriggs (talk) 04:43, 29 April 2015 (UTC)
Thank you. But note that it is not a suggested reference for the article. Just to what I was referring to on this talk page. YohanN7 (talk) 10:33, 16 June 2015 (UTC)

Quotes section[edit]

Why is there a quotes section? It seems to run afoul of WP:TRIV. Comparable articles don't have a quotes section. It doesn't impart any useful information about the subject to the reader. This section ought be removed. (talk) 19:08, 15 June 2015 (UTC)

The quotes section belongs because AC is kind of controversial and the quotes expresses some of the (sometimes) strong sentiments about it in a clever way that no dry technical section could convey to the average reader. The idea that quotes do not belong in WP is just nonsense. YohanN7 (talk) 13:49, 17 June 2015 (UTC)