Talk:Axiom of global choice

Formulation in the language of ZFC

"The axiom of global choice cannot be stated directly in the language of ZFC" this highly depends on the formulation of global choice. The very common version of "there is a class function $F:V\rightarrow V$ such that $\forall xF(x)\in x$" (whereby class means some formula with one free variable) is equivalent to the statement "there is a class well-ordering of the universe", which is equivalent to the statement "V=HOD", the last of which can be formulated in ZFC (for reference: see the chapter on HOD in Jech). --79.177.189.197 (talk) 18:54, 3 January 2016 (UTC)

Stronger?

On its face, this axiom is stronger than the axiom of choice. But Jech's book says that, in Bernays-Gödel, it doesn't prove any new statements that only mention sets. If that's true, then maybe "stronger" by itself is a bit confusing. Joeldl 16:42, 18 February 2007 (UTC)

I guess that it depends on whether you think that proper classes matter or not. JRSpriggs 06:35, 19 February 2007 (UTC)

Yes, that's right. Joeldl 12:22, 19 February 2007 (UTC)

What is this article about?

This article refers to the textbooks by Kelly and by Jech. But neither Kelly, nor Jech speak about "axiom of global choice". 7 months ago I initiated a discussion in Mathoverflow on what this term, "axiom of global choice", means: http://mathoverflow.net/questions/107650/axiom-of-global-choice. Up to now nobody gave an answer. Can anybody explain the necessity of this article in Wikipedia? Eozhik (talk) 04:02, 19 April 2013 (UTC)

I do not read Mathoverflow or other mathematics-related websites. If you want to discuss Wikipedia articles with me, you must do so on Wikipedia.

I have no idea why you would ask this question. The answer is clear on the face of the article. This is a stronger form of the axiom of choice, suitable for class theories like Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory. The ordinary axiom of choice is inadequate to handle proper classes of non-empty sets. If you can prove otherwise, then please do so here. JRSpriggs (talk) 10:45, 19 April 2013 (UTC)

Can you give references? I mean, sources where this "axiom of global choice" is described. Eozhik (talk) 18:04, 19 April 2013 (UTC)

James, if there are no references, then you should agree that there is a problem, isn't it? Eozhik (talk) 14:53, 20 April 2013 (UTC)

By the way, Kelly formulates the (usual) axiom of choice exactly for a proper class of non-empty sets, namely, for the class $U\setminus \{0\}$ of all non-empty sets (this is the axiom IX in his list). So these your words look also puzzling: "The ordinary axiom of choice is inadequate to handle proper classes of non-empty sets. If you can prove otherwise, then please do so here." Eozhik (talk) 15:39, 20 April 2013 (UTC)

Gentlemen, if there are no objections, I nominate this article for deletion. Eozhik (talk) 04:06, 21 April 2013 (UTC)

And ladies? GeorgeLouis (talk) 22:47, 21 April 2013 (UTC)

Apologies to ladies. Eozhik (talk) 08:22, 22 April 2013 (UTC)

Nomination of Axiom of global choice for deletion

A discussion is taking place as to whether the article Axiom of global choice is suitable for inclusion in Wikipedia according to Wikipedia's policies and guidelines or whether it should be deleted.

The article will be discussed at Wikipedia:Articles for deletion/Axiom of global choice until a consensus is reached, and anyone is welcome to contribute to the discussion. The nomination will explain the policies and guidelines which are of concern. The discussion focuses on high-quality evidence and our policies and guidelines.

Users may edit the article during the discussion, including to improve the article to address concerns raised in the discussion. However, do not remove the article-for-deletion notice from the top of the article. — Preceding unsigned comment added by Eozhik (talk • contribs) 04:49, 21 April 2013 (UTC)

The difference between: ordinary choice, 'weak' global choice, and 'strong' global choice

The ordinary form of the axiom of choice says

• Ordinary form: Every set of nonempty sets has a choice function.

The 'weak' form of the axiom of global choice says

• 'Weak' form: Every class of nonempty sets has a choice function.

The 'strong' form of the axiom of global choice says

• 'Strong' form: Every collection of nonempty classes has a choice function.

Perhaps these notions can be better understood by making them relative to V_κ instead of V:

Ordinary $\forall x\in V_{\kappa }\left[\emptyset \notin x\implies \exists f:x\rightarrow \bigcup x\quad \forall a\in x\,(f(a)\in a)\right]\,;$

'Weak' global $\forall x\in V_{\kappa +1}\left[\emptyset \notin x\implies \exists f:x\rightarrow \bigcup x\quad \forall a\in x\,(f(a)\in a)\right]\,;$

'Strong' global $\forall x\in V_{\kappa +2}\left[\emptyset \notin x\implies \exists f:x\rightarrow \bigcup x\quad \forall a\in x\,(f(a)\in a)\right]\,.$

In other words, the distinction lies in whether x ranges over V_κ or V_κ+1 or V_κ+2.

However, the 'weak' and 'strong' forms of global choice are only apparently different because the classes in the strong form can be cut down to the subclass of sets of minimal rank, and that subclass is a set. And then the collection of nonempty classes becomes a collection of nonempty sets, and any collection of sets is merely a class of sets. JRSpriggs (talk) 13:04, 21 April 2013 (UTC)

• A question: what do you mean by "collection of classes"? Eozhik (talk) 13:12, 21 April 2013 (UTC)

• And these notations -- V_κ, V_κ+1, V_κ+2 -- where are they from? Eozhik (talk) 13:20, 21 April 2013 (UTC)

Staying within the realm of class theory (where one has variables over classes), a collection {X|P(X,Y₁, ... Y_n)} of classes X could be understood as a predicate P with some class variables Y₁, ... Y_n as parameters with a free variable X over the classes which may or may not belong to the collection. See Von Neumann universe for your other question. JRSpriggs (talk) 13:38, 21 April 2013 (UTC)

First: "See Von Neumann universe for your other question" -- this doesn't sound polite. Eozhik (talk) 14:07, 21 April 2013 (UTC)

Second: From your reference I deduce that by "class" you mean not what Kelley or Gödel-Bernays mean by it (otherwise the meaning of the term "collection of classes" would remain unclear). If so, then I suppose, the article must contain a special section devoted to your interpretation of "axiom of global choice". Eozhik (talk) 14:07, 21 April 2013 (UTC)

The article no longer contains any mention of collections, so it is not necessary to explain them there. I merely mention it here so that you can understand why I spoke previously about 'weak' and 'strong' forms of the axiom of global choice.

If you want a clearer answer about the other matter, then you should make your question clearer and more polite. JRSpriggs (talk) 15:28, 21 April 2013 (UTC)

If you wanted me to understand your idea about 'weak' and 'strong' forms, then you didn't achieve the goal. However, I am glad that these words are eliminated. Eozhik (talk) 16:08, 21 April 2013 (UTC)

To Eozhik: I see no reason to think that I am using "class" differently than Kelley or Bernays. A set is a class which is an element of another class. A class could be thought of as a collection which is a member of another collection. You seem to think that there is some hard boundary beyond which one cannot further aggregate things, but there is no reason to assume that. As Inaccessible cardinal#Models and consistency says "If V is a standard model of ZFC and κ is an inaccessible in V, then: V_κ is one of the intended models of Zermelo–Fraenkel set theory; and Def (V_κ) is one of the intended models of Von Neumann–Bernays–Gödel set theory; and V_κ+1 is one of the intended models of Morse–Kelley set theory.". There is no reason not to consider also theories describing V_κ+2. In such a theory, elements of rank upto and including κ+1 would be called "collections"; elements of rank upto and including κ would be called "classes"; and elements of rank less than κ would be called "sets". Reading Reflection principle may make this all seem more plausible to you. JRSpriggs (talk) 07:59, 24 April 2013 (UTC)

I've neve seen a set of axioms corresponding to V_κ+2; it's possible that no reasonable set of axioms exist, unless you want to refer to second-order MK. But that is also problematic, as the relationships between the axioms of the metatheory and the axioms of set theory become complicated. Perhaps we should stick to (v)NBG and MK, and leave V_κ+2 to the metalogicians. — Arthur Rubin (talk) 16:54, 24 April 2013 (UTC)

I don't think about hard boundaries. What I am thinking about is that if one speaks about "collections of classes", he must give a system of axioms describing this term. That is the weak point in your explanations. By the way, you can join the discussion at MathOverflow which I initiated, http://mathoverflow.net/questions/107650/axiom-of-global-choice, I am sure, you'll find like-minded people there. Eozhik (talk) 22:19, 24 April 2013 (UTC)

Some of the large cardinal numbers listed at List of large cardinal properties are defined in ways that implicitly (or even explicitly) refer to models with multiple layers 'over the top' so to speak. These include: Superstrong cardinals, Extendible cardinals, Huge cardinals, and Rank-into-rank cardinals. JRSpriggs (talk) 01:42, 25 April 2013 (UTC)

Equivalences

Some of the equivalences are invalid in NBGU (von Neumann–Bernays–Gödel set theory with Urelements) or MKU (Morse–Kelley set theory with Urelements). Perhaps this should be mentioned. — Arthur Rubin (talk) 17:04, 21 April 2013 (UTC)

• All of the equivalences are valid in vNBGU or MKU plus US or UO, where:

  US is

  The class of urelements is a set, and

  UO is

  The class of urelements is either a set or equipollent to a class of ordinal numbers.

  (If the class or urelements is a set, it can be well-ordered by the axiom of choice, is equipollent to an ordinal number, and is hence equipollent to a class of ordinal numbers using the von Neumann definition of ordinal number.) In either case, the model of vNBGU or MKU can be simulated by the "pure sets" of that model, which models vNBG or MK, respectively.

• "The universe can be well-ordered" is a consequence of global choice in MKU, but not in NBGU.
• "The universe is equipollent to the class of ordinal numbers" is not a consequence of global choice in NBGU or MKU.

Unfortunately, I do not have a reference book available which discusses global choice in the presense of urelements. If I posted these facts with quasi-proofs on my web site, it would probably be allowable, per WP:SPS, but.... — Arthur Rubin (talk) 23:18, 24 April 2013 (UTC)

Scratch the assertion of the implication in MKU above. It appears not to follow. What follows is that if there is a formula which, for every non-empty class, chooses an element of it, then the universe can be well-ordered.

According to a published paper I cowrote, (Paul E. Howard; Arthur L. Rubin; Jean E. Rubin (December 1978). "Independence Results for Class Forms of the Axiom of Choice". Journal of Symbolic Logic. 43 (4): pp. 673–684. {{cite journal}}: |pages= has extra text (help)):

The forms presently in the article are:

• C2,
• Godel's E, (which is obviously equivalent to C2, see my parents' book, Equivalents of the Axiom of Choice, II, for which I provided a full citation somewhere in the archives of my user talk page),
• W4, and
• W2.

The paper goes on to prove that W2 (actually, W1, proved equivalent) → W4 → C2; the the latter implication is not reversable in NBGA (my NBGU), and that the former implication is not reversable in NBGA, assuming the existence of an inaccessible cardinal.

It also notes that in the presence of an axiom called RA,

$V=\bigcup _{\alpha \in On}A_{\alpha }$, where $A_{\alpha }$ are sets,

then all of the forms mentioned in that paper are equivalent. RA follows immediately from my US, and almost immediately from my UO above, using the von Neumann hierarchy.

I'm not sure how to incorporate this into the article. I should add that, since any model of MK(U) is a model of NBG(U), any implications shown in NBGU hold in MKU, but not necessarily the reverse. — Arthur Rubin (talk) 00:17, 25 April 2013 (UTC)

Category:Axiom of choice?

It is true that "strictly speaking this is not an equivalent of the axiom of choice nor a weaker forms of that principle. thus it does not satisfy the criterion for the subcate", as long as that category is defined as "for equivalents of the axiom of choice, and weaker forms of that principle." However, why not redefine that category as "for the axiom(s) of global choice, and weaker forms"? This sounds logical to me. Boris Tsirelson (talk) 19:49, 18 December 2016 (UTC)