Talk:Von Neumann universe

What about urelements?

Does that mean that every element of every set is always a set itself, too? Are sets like 'the set of current or former U.S. Presidents', whose elements aren't sets, disallowed? If so, what the f*** does it mean "The band Metallica has four members"? The most obvious answer is that the sets {James Hetfield; Lars Ulrich; Kirk Hammett; Robert Trujillo} and 4 = {0; 1; 2; 3} = {{}; {{}}; {{}; {{}}}; {{}; {{}}; {{}; {{}}}}} are equipotent. But here I learn that the former is not a set, as its elements aren't themselves sets, so what does it mean?

In the Von Neumann universe, if a set has any elements at all, then the elements must themselves be sets. "Sets" containing elements which are not sets are not allowed in this universe by definition. What you are probably looking for are universes with urelements, which allow sets containing members that are not themselves sets. - Gauge 05:46, 24 Jan 2005 (UTC)

Assume ZF^- with the necessary modifications to allow urelements. Following the links between transitive sets, von Neumann universe and well-founded sets it seems that well-founded sets can have urelements in their transitive closure. However, it is not clear whether von Neumann universe

1. is the class of all well-founded sets whose transitive closures do not contain urelements (so that it is the union of all stages in the von Neumann hierarcy.)

2. is the class of all well-founded sets, even if their transitive closures contain urelements (this seems to be the most honest interpretation of the introduction of this article.)

3. is the class of all well-founded sets and all urelements (so that only ill-founded sets are excluded.)

I would be grateful for a clarification of this matter. Lapasotka (talk) 20:37, 17 March 2011 (UTC)

It's your number 1. --Trovatore (talk) 21:05, 17 March 2011 (UTC)

Thank you for the quick response! I added this important piece of information into the definition. Lapasotka (talk) 21:31, 17 March 2011 (UTC)

The point is to see what modifications are necessary to add urelements. You could define urelements as sets with no members. Then you must not only drop the axom of Foundation but also change the axiom of extensionality since all urelements have the same members (i.e. none) yet they are distinct. Then you are not in Zermelo set theory at all and you will have to change the definition of the von Neumann universe accordingly---by excluding urelements. Or you could say each urelement is its own sole member, and keep all the axioms of Zermelo Fraenkel except Foundation. Then you can keep the usual definition of the von Neumann universe since these urelements are plainly not well-founded sets.Colin McLarty (talk) 13:35, 11 April 2011 (UTC)

Define rank

The page "rank (set theory)" redirects to this page. Therefore, this page should define rank without using the word rank in the definition. For example:

"The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set."

This defines rank as a number greater than other ranks. If a person is looking for the definition of a rank, this circular dependency will keep them from finding it. -- kainaw™ 19:22, 2 September 2011 (UTC)

The definition is actually correct; the circularity is apparent rather than real. For example the empty set has no members, so every ordinal is greater than the ranks of all members of the empty set. So the rank of the empty set is the least ordinal, period, which is zero. You proceed from there.

Just the same, I agree that it's not too friendly to make the reader work through that problem, and it's also not the most useful definition in practice. I propose replacing it with something like "the rank of x is the least ordinal α such that $x\in V_{\alpha +1}$". --Trovatore (talk) 23:23, 2 September 2011 (UTC)

The last sentence in the Von Neumann universe#Definition section says "The rank of a set S is the smallest α such that $S\subseteq V_{\alpha }\,.$". If you ignore the first sentence and a half of that section, it gives a non-circular definition. JRSpriggs (talk) 03:18, 3 September 2011 (UTC)

Right, and for that matter the apparently circular definition is not actually circular anyway. Just the same, I agree with Kainaw that it's a problem. I'm not sure of the best fix; it probably requires rewording the lead entirely.

I would prefer that the lead be changed to emphasize the iterative, rather than recursive, nature of the hierarchy. I know that's not entirely non-controversial; fans of John Conway are likely to feel otherwise. However I do feel it's more reflective of the way set theorists usually conceive of these things. --Trovatore (talk) 03:44, 3 September 2011 (UTC)

Could this image be used as a example?

Isn't von Neumann universe some sort of bounding volume hierarchy? --Pasixxxx (talk) 18:19, 3 December 2011 (UTC)

No. JRSpriggs (talk) 00:52, 5 December 2011 (UTC)

V and ZFC

I am a little confused at the momemt, and it's sometimes embarrassing to reveal how confused you really are. However, reading the section "Philosofical Perspectives" made me feel a little better. Pretty much everyone is confused. At least, not everyone agrees with everybody else. Good!

I initially wrote a longer post, but decided not to actually post it to save me some of the embarrassment. Basically, I would like to see a longer article. As it stands it's pretty much like a good riddle. It is possible, but pretty time consuming, to find the correct answers.

Here is one "anwer" that I now formulate as a question: Could V in some sense be called the intended model for ZFC? If so, in which sense?

It's hard to think of elements of V that aren't sets (according to ZFC) and, vice versa, permissible sets (in ZFC) that aren't in V. Quote from the article: "In ZFC every set is in V." I think so too, but this isn't obvious at all. But elements of V seem all to be valid ZFC sets by construction. Things like real numbers can be built from the natural numbers. [I am thinking reals as equivalence classes of Cauchy sequences of rational numbers here.] They ought to appear pretty early on in the hierarchy, but when exactly? YohanN7 (talk) 18:48, 4 May 2012 (UTC)

Yes, V is the "intended model" of ZFC, with the only caveat being that (from a realist perspective) V cannot be a completed totality, an actual individual object (if it were, it would have to be a set). Therefore the statement "V is the intended model" has to be understood as shorthand for a longer-winded claim, one that would describe the intended interpretation of statements in the language of ZFC, with quantifiers ranging over objects that satisfy "I am in V" as a predicate. --Trovatore (talk) 22:29, 4 May 2012 (UTC)

Ok. Quote from the article: If κ is an inaccessible cardinal, then V_κ is a model of Zermelo-Fraenkel set theory (ZFC). Here V_κ is a set. In order to prove that it is a model of ZFC, one would still have to produce that longer-winded claim describing the interpretation of ZFC statements? Then one goes about to show that every axiom of ZFC is true in V_κ. Is that the order of buisiness?

Example. The axiom of power set holds in V_κ. If $S\in V_{\kappa }$ then $rank(S)=\lambda <\kappa$. The power set will have rank $\lambda +1<\kappa$. (Hm. For this axiom to hold it seems to suffice that $\kappa$ is a limit ordinal.)

Would the fact that V is a class and V_κ a set make a substantial difference on the practical level?

I'd still like a longer article since V seems to be very important (= used in many contexts). YohanN7 (talk) 11:03, 5 May 2012 (UTC)

I had a look at L. That article parallels a little bit what I'd like to see here. There are things in the article universe too that could well be here. Especially that one can use the axiom of regularity to show that (ZFC) sets not in V don't exist. Also, we cannot actually prove that the axiom of choice holds in V (without actually assuming it), can we? So saying that V is a model of ZFC is incorrect strictly speaking. But V can still be "the intended model" of ZFC as V very well might be a model of ZFC, and a very nice model at that. It is a model of ZF according to the article about L. Do I make sense? YohanN7 (talk) 12:41, 5 May 2012 (UTC)

If κ is a limit ordinal greater than ω, then V_κ will satisfy all the axioms of ZFC except possibly some instances of the axiom schema of replacement.

Given a set S and a relation E⊆S×S, then the statement that (S,E) is a model which satisfies a sentence φ (in the language of set theory) can be encoded in set theory as a predicate with two free variables (one for the model and one for the sentence). Thus it is possible to encode statements and arguments about ZFC (its meta-theory) as statements and deductions within ZFC insofaras they apply to that particular model. JRSpriggs (talk) 20:55, 6 May 2012 (UTC)

Zermelo was the true author of the so-called von Neumann universe

Zermelo was the real author of the so-called von Neumann universe. See Gregory H. Moore, "Zermelo's axiom of choice", Dover Publications, 1982, 2013, pages 270, 279. See also Ernst Zermelo, "Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre", Fundamenta Mathematicae, 16 (1930) pages 36–40. Would anyone like to add a note to the wiki page to explain that the von Neumann universe was first published by Zermelo? I have been reading the relevant articles by von Neumann himself in the 1920s. Those articles obviously mention the von Neumann construction for the ordinal numbers, which he did in fact first publish, but there is no sign of the so-called von Neumann universe in his papers. The closest thing to the von Neumann universe published by von Neumann himself is in the 1928 article "Die Axiomatisierung der Mengenlehre", Math. Zeit. 27 (1928) pages 235–237, but that is a model which looks very much like the Gödel style of prime-number model, not at all like the von Neumann universe.
Alan U. Kennington (talk) 14:11, 6 August 2013 (UTC)

So apparently von Neumann did not develop the VNU whereas Zermelo did. Did anyone other than Moore notice this? Tkuvho (talk) 14:21, 6 August 2013 (UTC)

Well, I noticed it when I read through all of the articles where von Neumann could have presented the von Neumann universe in the 1920s. I could find no sign of it. And then finally, while reading through my books on the subject, I noticed that Moore said this on page 270 of the above-mentioned book.

Von Neumann obtained his inner model Π as the image of a function ψ defined by transfinite recursion on the class of all ordinals, but did not use the cumulative type hierarchy often attributed to him.

By "cumulative type hierarchy", Moore means of course the von Neumann universe. Moore gives the reference von Neumann 1929, pages 236–238, for the inner model Π. ("Über eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre", J. Reine Ang. Math 160 (1929) 227–241. I've just skimmed through this article and saw no sign of the von Neumann universe at all. But it is definitely in the 1930 Zermelo paper as Moore indicates.) Moore says on page 270:

To obtain these theorems, Zermelo introduced what is now called the cumulative type hierarchy for set theory.

This is the Zermlo paper which I referred to above, which does definitely present the so-called von Neumann universe.
Alan U. Kennington (talk) 15:47, 6 August 2013 (UTC) Alan U. Kennington (talk) 16:11, 6 August 2013 (UTC)

Redirect from "V-Hierarchy"

This needs a redirect from "V-Hierarchy". I couldn't find it on my own without that.--208.72.139.94 (talk) 04:46, 23 February 2014 (UTC)