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)

What is Zermelo Fraenkel good for, anyway? Just about everything in any useful form of mathematics is built from urelements. Even the ZF "interpretation" of positive integers is useless, because there are many contexts in which sets of numbers are considered where the "license to reduce" e.g., {0; 1; 2; 3} to 4 doesn't make sense.

198.228.228.153 (talk) 13:14, 29 June 2014 (UTC)Collin237


 * To 198.228.228.153: This is not a problem which exclusively affects ZFC. In most mathematical models of physical systems, the meaning of an element of the model depends on its context, that is, on how it is used in the interpretation of the model &mdash; which variable is assigned that element as its value. Whether an element is interpreted as a single integer or as a set of integers depends on the meaning attached to the variable which is assigned that value. JRSpriggs (talk) 05:46, 30 June 2014 (UTC)

The role of ZF and the von Neumann universe in mathematics
To 198.228.228.153: It seems to me that ZF set theory is only a testing ground for mathematical theories. If you can define a model for a theory within ZF, you get all the advantages of the last 100 years' research into consistency, completeness, independence, and so forth. ZF has two big advantages. It has been very well researched, and it is capable of providing models for most mathematical theories. It's solid and it's big. If you can build a model for a theory inside ZF, it's probably as self-consistent and "valid" as you can get at this point in history.

When you have demonstrated that you can build a model for a theory within ZF, you can throw away that interpretation and return to the theory without any particular ZF-based model. For example, you might first build a model for the non-negative integers as 0=∅, 1={∅}, and so forth. In this framework, you can validate your axioms, and then you can just forget the particular representation. Similarly, one represents the real numbers as Cauchy sequences or Dedekind cuts to verify that it's all self-consistent and has the right sorts of properties. Then you throw away the representation when it has served its validation purpose.

If you don't believe that the representation is thrown away, ask 10 typical mathematicians if they believe that the propositions $$1\in2$$, $$1\subsetneq2$$, $$1\cup3=3$$, and $$\cap\, 9=0$$ are true? Chances are that 9 out of 10 of them won't think these statements have any meaning at all. That's because most mathematicians are not interested in ZF models. They know that someone has validated the theories they use, and they just get on with their mathematical lives. Similarly, no one really believes that the number $$\pi$$ is an equivalence class of Cauchy sequences of rational numbers.

The original plan of Peano, Frege, Whitehead and Russell, Hilbert, and others, was to show that mathematics can be modelled within some kind of axiomatic system where consistency can be guaranteed in some way. That plan didn't 100% succeed, but a huge amount of clarity was brought to the issue during the attempt. Some disturbing surprises turned up too, but overall, ZF does seem fairly solid, and it has survived for 100 years, roughly speaking. So it's probably okay. And Gödel showed that AC and GCH are consistent with ZF, and Paul Cohen showed that AC and GCH are independent of ZF. Since 1963, the most important questions have been resolved to most people's satisfaction. The remaining little bits and pieces of the picture are still studied in model theory, but all in all, ZF (with or without AC and/or GCH etc.) looks like it's safe to use. Most importantly, it's a framework where you can define models for theories, and can be fairly confident that if your theory can be embedded/immersed/interpreted in ZF set theory (e.g. in the von Neumann universe), then probably your theory has no consistency problems, assuming that the interpretation map from the theory to the model is validly defined.

In other words, what I'm trying to say is that mathematics does not reside inside ZF. So the atoms don't matter. All that matters is whether you can find a map of some kind from the objects of a theory into ZF. It doesn't matter if a potato is modelled as the set ω and a tomato as the set ω × ω. If the theory is valid when mapped into ZF, you've got the benefit of some kind of standard validation. It's a kind of benchmark which everybody feels they understand well enough. And if ZF does have serious problems which are not yet discovered, we're all in trouble. You'll probably get a prize for discovering any contradictions in ZF. So it's win-win. Either ZF is safe to use, or you get fame and a big prize.

Conclusion: ZF is just a testing ground for theories. Mathematics doesn't reside inside ZF. --Alan U. Kennington (talk) 13:41, 30 June 2014 (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. -- k a i n a w &trade; 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 &alpha; such that $$x\in V_{\alpha+1}$$".  --Trovatore (talk) 23:23, 2 September 2011 (UTC)
 * The last sentence in the Von Neumann universe 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 Vκ is a model of Zermelo-Fraenkel set theory (ZFC). Here Vκ 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 Vκ. Is that the order of buisiness?


 * Example. The axiom of power set holds in Vκ. 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 Vκ 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 Vκ 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&sube;S&times;S, then the statement that (S,E) is a model which satisfies a sentence &phi; (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 &Pi; as the image of a function &psi; 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 &Pi;. ("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)

Of course, the Neumann sources cited in the article do not contain the hierarchy, but another source does: John von Neumann: ‚‚Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre‘‘, 1928, in: ‚‚Journal für die reine und angewandte Mathematik‘‘ 160 (1929) 227–241. There p. 236f the cumulative hierarchy (nameless). Therefore Neumann is the true author and not Zermelo. Please correct! Wilfried Neumaier (talk) 18:44, 27 December 2022 (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)
 * Well, you're welcome to create it if you like. You might have to register an account to do it.  You probably should also do a little searching to see if there's an alternative meaning for the term, so as not to send people to the wrong place.  --Trovatore (talk) 07:11, 23 February 2014 (UTC)

Philosophical perspectives
The section "Philosophical perspectives" is marked as not containing any references. Well, I just tried to think of where I might find supporting references amongst my collection of 45 books on set theory and foundations of mathematics, and it seems to me that none of them will support this kind of philosophical summary. In fact, I don't think that section makes any sense at all, and it doesn't agree with anything that I have read. It's more like an afternoon tea discussion amongst the undergrads trying to figure out stuff. Sorry to be so negative about it. What I'm trying to say is that either that section should be deleted or improved. I think it's kind of embarrassing as it stands because it says nothing. It will just confuse people who are unfamiliar with the subject and annoy people who understand it.

My personal preference would be to simply delete it. I don't think it can be repaired. Then if someone wants to add something about interpretation as a new section, they could do that by starting with some authoritative literature as a source of references. — Preceding unsigned comment added by Alan U. Kennington (talk • contribs) 02:49, 7 June 2014 (UTC)

Moving the philosophical perspectives to the discussion page
I'm moving the unreferenced philosophical perspectives section to this disccusion/talk page. I think it's dubious that the statements there can be backed up by book references or by wikipedia cross-references. They seem to be someone's personal comments on the possible range of philosophical interpretations, but there is not much value in it as far as I can see. It's not as if V is widely discussed in the literature for its philosophical significance any more than any other model for ZF or ZFC or other theories is discussed. V is just one special case of a model amongst hundreds. (I am retaining the original "unreferenced section" notification.) --Alan U. Kennington (talk) 07:37, 13 June 2014 (UTC)
 * Um. I wrote that, long ago.  I can't really object to your point that it's unreferenced.  However, "just one case of a model" is completely wrong.  V is the intended model.
 * I think everything in the section is accurate, and I think it is valuable. But you're quite correct that it's necessary to source it.  If I get a chance I'll look around. --Trovatore (talk) 10:29, 13 June 2014 (UTC)
 * One good place to look would be Stewart Shapiro's Foundations Without Foundationalism. --Trovatore (talk) 10:31, 13 June 2014 (UTC)
 * Also Penelope Maddy, Believing the Axioms, I. --Trovatore (talk) 10:40, 13 June 2014 (UTC)

Of course, I didn't mean my comments as a criticism of the author of that text. It just bothers me that someone or other slapped an "unreferenced section" label on that section, and I think that this is an obstacle to the article getting a higher ranking. "C class" doesn't sound very good to me, for such a core topic as the von Neumann hierarchy.

I've noticed that in the German version of this page (which is definitely not a translation of the English version), they have an applications section. That might be a good idea for the English-language version. So I added a section for that. I think the philosophical aspects of the interpretation of the von Neumann universe could go in such an "applications" section.

In my opinion, there needs to be some explanation of the status of the model. After all, ZF is often used as a source of models in an "underlying set theory" for more general model theory, particularly for a wide range of independence and relative consistency proofs. Therefore clarification of the status of V is at the core of a lot of model theory. I think this is a strong motivation to provide some philosophical background for V. After all, it seems to be a kind of boot-strap model from which other models are derived. It's a very circular thing, building V within ZF and then claiming that V is a model for ZF. It is this circularity which requires some clarification. It's related, I think, to the fact that Gödel's incompleteness theorem says you can't prove consistency for ZF within ZF (since ZF contains the integers). I think any philosophical commentary should connect the status of V with Gödel's results. --Alan U. Kennington (talk) 10:54, 13 June 2014 (UTC)
 * It appears to be a circularity when you present both views at once, yes. But that's because you're trying to affirm two incompatible views at the same time.  For the formalist, ZF comes first and V follows; for the realist, it's the other way around.  Very roughly speaking.
 * So yes, this does require clarification. That's kind of what I was trying to do in the passage you removed.  You were probably correct to remove it, because of the sourcing issue, but the basic content ought to be restored somehow. --Trovatore (talk) 10:59, 13 June 2014 (UTC)

Agreed. In fact, I think the question of the status of V needs several paragraphs of explanation. So you say, you can resolve the circularity by starting with one or the other as "a-priori", and then the other derives some status from it. This could be explained within the very general context of models for theories though. As you say, it is a question of formalism versus realism. So it really leads into the big questions of the direction mathematical logic took in the 20th century. In my personal opinion, it was the crisis created by Gödel which made everyone realise that Hilbert's program wasn't going to work. And that led to the split between proof theory and model theory. Nowadays, I would say that mathematical logic takes the very cynical view that languages are the primary thing, and theories are completely arbitrary, and models are an afterthought. I take the opposite view. I think that mathematical objects exist in human minds and in the mathematical community. The theories exist only to try to axiomatize and systematize large bodies of objects whose existence is self-evident to mathematicians. As someone once observed, the paradoxes in mathematics never put the brakes on mathematics. They just don't matter because they are in the realm of the theory, which is an imperfect description of the reality. The reality continues to exist when the theory hits a crisis.

In other words, what I'm saying is that any references for this subject could be really references for the general issues of models versus theories. That's just my two cents' worth. I'm not an expert in this area. I just want to understand what it all means! --Alan U. Kennington (talk) 11:11, 13 June 2014 (UTC)


 * The above comments remind me very much of a similar problem with the interpretation of quantum mechanics. Every few years, someone takes a poll of attendees leaving some conference, as to which interpretation they prefer, and, over the course of decades, you can see overwhelming, complete shifts of viewpoints. One also sees large differences between the opinions of the "average practicing physicist" and that subgroup that "bothered to think about the problem in depth", and these two tend to be in diametric opposition.  So the description and determination of the beliefs is a sociological, anthropological problem -- an interesting one, but where things shift around.  Alan Kennington seems to be a platonist, given the remarks above (e.g. "The reality continues to exist when the theory hits a crisis"), but by reading through some of the "13 essays in the philosophy of mathematics" indicates there is much more to it than just that -- platonism itself, the belief in the reality of mathematics, is just another viewpoint, and is hardly an "obviously correct" viewpoint (except to those who believe in it, in which case it appears to be self-evident. How odd...)


 * That said, yes, a good discussion of the different philosophical viewpoints is worthwhile, even if it is unreferenced. If not here, then somewhere? 84.15.181.100 (talk) 08:20, 13 June 2016 (UTC)

Existential status of V
I have added a "stub subsection" for the existential status of V. I think (hopefully) that this is more or less the same as the topic of the "philosophical perspectives" section which I (so rudely) removed. I have tried to add some references for it.

If this new "stub subsection" is unacceptable, please modify or delete it. --Alan U. Kennington (talk) 12:58, 13 June 2014 (UTC)

Philosophical perspectives (moved out of the main page)
There are two approaches to understanding the relationship of the von Neumann universe V to ZFC (along with many variations of each approach, and shadings between them). Roughly, formalists will tend to view V as something that flows from the ZFC axioms (for example, ZFC proves that every set is in V). On the other hand, realists are more likely to see the von Neumann hierarchy as something directly accessible to the intuition, and the axioms of ZFC as propositions for whose truth in V we can give direct intuitive arguments in natural language. A possible middle position is that the mental picture of the von Neumann hierarchy provides the ZFC axioms with a motivation (so that they are not arbitrary), but does not necessarily describe objects with real existence.

Is V really a model for ZFC? Where is C shown to be true?
I am skeptical about the claim that the full von Neumann hierarchy V is a model for ZFC. I think this assertion arises from the confusion regarding the meaning of "Zermelo-Fraenkel set theory". Some people define it as ZF and other define it as ZFC. I know that AC was part of the original set of axioms historically, but that's not the point. The important point is that most authors (and I could give many examples) mean ZF without choice when they say "Zermelo-Fraenkel". But that is not the question I want to discuss here. It is only the background which could lead to a false or misleading assertion here.

If someone could provide a reference where there is a proof that V is a model for ZFC, that would be very useful. The Paul Cohen "Set theory and the continuum hypothesis" book says that V is a ZF model, and on page 88 says that AC and GCH are provable in ZF if you assume V=L. Well, that looks a fairly authoritative statement that the von Neumann universe on its own does not give you AC. At least it is not provable! Smullyan and Fitting, "Set theory and the continuum problem", on page 97, say that V is a model for ZF without any mention of AC. According to Gregory H. Moore, "Zermelo's axiom of choice", page 281, Gödel showed in 1938 that AC is satisfied by VΩ, where Ω is the least strongly inaccessible ordinal. I don't want to do a complete literature search on this. Perhaps the full-time professionals in this area can throw some light on it.

Just to recapitulate, my question is whether it is provable that the complete hierarchy V satisfies AC. If so, could someone provide a reference. If not, I think the wiki page should be modified to say that V is a model for ZF, not for ZFC. --Alan U. Kennington (talk) 03:39, 7 June 2014 (UTC)


 * This whole construction does not make sense unless one assumes some underlying theory of sets. If that theory is ZFC, then V&kappa; (for &kappa; inaccessible) will be another model of ZFC because the choice function for any set of nonempty sets in V&kappa; will also be in V&kappa;. See the beginning of the second paragraph of Inaccessible cardinal for a possibly better wording.
 * If the underlying theory is merely ZF, then one must ask what does "inaccessible cardinal" mean? If it is a strong limit cardinal (among other things), then for all κ < λ, 2κ < λ. If the powerset of any ordinal number below λ can be injected into λ, then the axiom of choice does hold in Vλ. JRSpriggs (talk) 05:24, 7 June 2014 (UTC)

Reading between the lines here, I gather that there is no proof that AC is valid for the full von Neumann hierarchy V. But many sources assert that V is a model for ZF without assuming AC. I don't see that an understanding of inaccessible cardinals is relevant to asserting that V satisfies ZF, with or without AC. I have not seen it written anywhere that the definition of V requires the definition of inaccessible cardinals. When you say "does not make sense", do you mean that V cannot be defined without the axiom of choice. Suppose no one had ever discussed inaccessible cardinals of the axiom of choice ever. Then would V make sense? I think the answer is yes. It seems to me that V is defined to be closed under all of the constructive operations of ZF. The axiom of choice is very definitely not a constructive operation! So how can it be said that AC is required for defining V? I don't think that inaccessible cardinals play any role in the definition of the closure of V under all ZF operations.

I would like to separate out the downstream applications and analysis of V from the definition of it. To me, it seems that AC is not part of the definition. It is either a provable property or it is not. Inaccessible cardinals are likewise attributes of V, not part of the definition. And as far as I can see, AC cannot be proved for the full hierarchy V. Therefore it should not be asserted to be so on the wiki page. If you have a proof of AC for V, I think it should be referenced. If you need extra definitions, like inaccessible cardinals, in order to make the AC assertion, then that is a kind of conditional proof, requiring extra assumptions. Without those extra assumptions, it seems like V is not a ZFC model. --Alan U. Kennington (talk) 05:40, 7 June 2014 (UTC)

To put it another way, is the following statement true?

1. The von Neumann cumulative hierarchy V is a model for ZF. 2. Under certain additional assumptions [... insert assumptions here ...], it can be shown that V is a model for ZFC. --Alan U. Kennington (talk) 06:36, 7 June 2014 (UTC)
 * You can't prove that V satisfies choice without assuming choice. But this is nothing special about choice.  You also can't prove that V satisfies, say, replacement, without assuming replacement. --Trovatore (talk) 21:42, 7 June 2014 (UTC)

Thanks. That sounds like what I suspected. So I'm guessing that if the underlying set theory for defining the cumulative hierarchy V is ZF set theory, then V satisfies ZF (although not provably so within ZF), and if the UST is assumed to be ZFC, one may obtain a cumulative hierarchy V which satisfies ZFC.

So now another question which is directly relevant to the correctness of the article is whether the von Neumann universes defined with ZF and ZFC as the UST are different universes. If so, then I also ask myself what happens with the 158 choice axioms and choice-like principles in the Howard/Rubin book "Consequences of the axioms of choice", pages 91–103, about 80 of which are more or less independent. Since there are so many choices for the axiom of choice, if your UST determines what you get for the minimum cumulative hierachy, that means that the word "the" in the article needs to be replaced with "a". E.g., are the ZF, ZF+AC and ZF+ACω versions of V different?

My personal guess is that there is one and only one minimum cumulative hierarchy of sets (without atoms) which is closed under all of the operations of ZF. My guess is that the various axioms of choice have no influence on the construction at all. And I guess also that since it is not known, and not knowable, whether V satisfies the weakest AC, and the strongest AC, and everything in between, anyone is free to speculate as they wish. And if they speculate that V satisfies the full AC, they get certain consequences, which lead to the towers of metatheorems which I have read some of in the books on model theory. And if they don't speculate about V satisfying any AC axioms at all, they get nothing. And I think that is maybe why people add AC into the UST, namely to generate interesting theorems which are contingent on the ad-hoc assumption that the "construction" V satisfies some kinds of AC conditions.

I conclude from this that the article should be expressed in terms of ZF without AC, because V is really constructed from ZF alone, since AC is in no way a constructive axiom. (Similarly the axiom of regularity plays no concrete role in the construction.) This is not just fairly clear to a non-specialist like me. It also appears in the standard textbooks which I have consulted. I think the addition of C to ZF is perhaps just a habit because so many researchers need it to bootstrap their metatheorems and research. But that is no reason to add it everywhere, like pouring tomato ketchup over everything on the plate. But in my own work, I am trying to determine what can be done with no AC at all. (Yes, I realise that many authors have written extensively on this already.) My thinking is, if AC is superfluous, don't add it. Some foods don't need ketchup!

If I am right about any of this, I think some changes would be beneficial to the article. --Alan U. Kennington (talk) 03:28, 8 June 2014 (UTC)


 * If you are worried about constructiveness, then look at Kripke–Platek set theory. AC is not the only nonconstructive thing in ZFC.
 * The way that you are thinking about this makes no sense &mdash; there is at most one V.
 * A formalist would work within some theory of sets and then find that the V constructed using that theory satisfies that theory (duh), except for non-well-founded elements. He could choose to use a different theory, but there would be no question of comparing the Vs of these theories since they do not really exist.
 * A Platonist would believe that the real V already satisfies some theory of sets, and one cannot change that theory, so there are no other Vs to compare with the real one. JRSpriggs (talk) 11:14, 8 June 2014 (UTC)

Many thanks. That confirms my initial tentative understanding of the subject. Now it is no longer so tentative. As far as I can see, there is no possibility of proving that V is a model for ZFC, and the axiom of choice has really very little to do with the definition of V at all. I just personally prefer to see definitions and their properties clearly delineated. The article starts off talking about ZFC, which really is not the right context, I think. It's really a ZF concept which can be studied downstream from the definition within the context of AC questions. Hence I would prefer to see the article steer well clear of AC and the "C" in ZFC until after V has been defined. Then the implications for AC could be gone into.

By the way, I read through the original 1930 German article by Zermelo where von Neumann's cumulative hierarchy for ordinal numbers was first applied to a ZF universe of sets in a published work, and Zermelo explicitly excludes axiom of choice from any role in the definition of the hierarchy. He lays out the ZF axioms one by one, and says that AC is intentionally not one of them! --Alan U. Kennington (talk) 12:40, 8 June 2014 (UTC)
 * I think you're kind of missing the point. V is not a "ZF-concept".  It's a concept that can be directly understood without reference to any formal theory at all.  It's the same thing, whether you then choose to prove theorems about it using ZFC, ZF, Z, KP, ZFC+large cardinals, or what have you.
 * There is no reason to call out AC separately here as though it has some weird status different from those of other ZFC axioms. --Trovatore (talk) 20:20, 9 June 2014 (UTC)

That is in fact what I had assumed for many years until I looked at this wikipedia article. The section "V and set theory" says this:


 * If κ is an inaccessible cardinal, then Vκ is a model of Zermelo-Fraenkel set theory (ZFC) itself, ....

Now I had thought that the very definition of V was that it was the minimum model which is closed under the ZF "constructive axioms", particularly the infinity axiom, the power-set axiom, the union axiom and the replacement axiom, in the same sense that one defines the span of a set of vectors as the intersection of all vectors spaces containing a given set of vectors and closed under vector space operations. Also the same thing for the subgroup generated by a set of elements, and so forth for every algebraic category. So I thought that in the case of set theory, the von Neumann universe was the "span" of the empty set, and with ZFA, you get the span of a set of atoms, and so forth. So then it comes as a surprise that Vκ is a model for ZF, when supposedly Z is the minimum model closed under ZF operations. This is what triggered my fear that maybe when you "construct" V under ZFC assumptions, there could be some magical way that I don't know about whereby now a proper sub-model of V now becomes the minimum model for ZF. This is not just my own personal interest. I am sure that anyone who knows the basic "span" constructions of the algebraic categories would be scratching their heads too. So my question is: How come V is the span of the empty set in ZF theory, and then in ZFC theory, Vκ is the span of the empty set? It is the most basic thing one could ask about the von Neumann universe. Is it really the minimum model for ZF? Or is the minimum dependent on which additional axioms one throws in?

On the subject of minimality, I think one of the other very fundamental questions might ask with a bit of basic background in basic algebra is what is the relation to the constructible universe? It appears on the face of that L is also the minimum model for ZF because it contains the empty set and is closed under all constructive operations of ZF. This question seems to be a bit metaphysical to me because we can't ever "know" what the sets look like inside V ∖ L since they are not constructible. So the existence of sets in V ∖ L is a matter of "Platonic faith" in the same way that the existence of a well-ordering of the real numbers is a matter of faith in V, whereas a well-ordering of the real numbers in L can in principle be written down explicitly. If there is nothing "out there" beyond L, then maybe Lebesgue non-measurable sets don't "exist", for example. Once again, I think these questions are basic, not esoteric.

Q1: If V is the minimum model for ZF, then how come the sub-model Vκ is a model for ZF?

Q2: If V is the minimum model for ZF, then how come L is the minimum model for ZF?

Even someone with basic undergraduate algebra would ask such questions. And I still claim that AC should be introduced in the article after V has been defined. My apologies for persisting with this matter. --Alan U. Kennington (talk) 00:33, 10 June 2014 (UTC)
 * No no no. V is not a minimal model.  It's more like a maximal one.  At every stage, you take all subsets of the previous stage, not just enough to make separation true. --Trovatore (talk) 05:11, 10 June 2014 (UTC)

Okay. So one might think of L as minimal, and V as maximal. Then if AC is assumed to be in force, one can identify a stage Vκ where the transfinite recursion can be safely terminated while still ensuring closure under all ZF operations. I think that makes better sense now. Wouldn't you agree that the addition of some kind of note to this effect would increase the value of the page for the non-specialist reader? I don't want to add anything myself that I'm not rock-solid certain about. (I have only learned about this subject by reading books!) --Alan U. Kennington (talk) 05:46, 10 June 2014 (UTC)

Adequacy of V&omega;+&omega;
According to Von Neumann universe, "it is questionable whether such a universe is adequate for all 'ordinary mathematics'". However, from the compactness theorem and downward Löwenheim–Skolem theorem, we know that every consistent theory (which is countable as any actual theory must be) has a countable model. In particular, ZFC itself has countable models even if one adds additional large cardinal axioms (as long as the resulting theory is still consistent). If a model is countable, then one can map its elements injectively into V&omega;. Any n-ary relation on that base set may be represented by a set of ordered n-tuples of such elements. These n-tuples are also elements of V&omega;. To represent the whole model itself, one tags the elements of each relation with a symbol representing that relation. A tagged element is just an ordered pair of the symbol and the n-tuple and thus also an element of V&omega;. Then the set of all such tagged elements (a subset of V&omega; and thus an element of V&omega;+1) represents the whole model.

Since all such models are in V&omega;+1, there is plenty of room in V&omega;+&omega; to develop relationships between models and so forth. Thus V&omega;+&omega; can only be regarded as "inadequate", if one insists on models which satisfy some conditions from second-order logic or that the "actual" element relation be used in a model of set theory (i.e. a "standard" or "transitive" model) rather than a set of ordered pairs. JRSpriggs (talk) 13:51, 13 June 2014 (UTC)


 * Good! I hope you will add words to that effect to that subsection. Relating V to normal mathematics would make the article much more relevant to non-specialist readers. I have just broken up that subsection into 3 paragraphs which are really about different subjects. I think it's very important that most normal mathematics can be done in Vω+ω. A lot of people would say that Vω+3 is adequate. Maybe the "Applications of V as models for set theories" subsection should be at section level, and then the cases Vω, Vω+ω, etc., could be at subsection level.

--Alan U. Kennington (talk) 14:13, 13 June 2014 (UTC)

&omega;+&omega; itself
(However, since the fairly "ordinary" set ω+ω is not itself an element of Vω+ω, it is questionable whether such a universe is adequate for all "ordinary mathematics".) []

That's a concern for "ordinary mathematics" is it? This tacitly assumes the Von Neumann representation of the ordinals. I notice that the article defines


 * $$V_\alpha := \bigcup_{\beta < \alpha} \mathcal{P} (V_\beta)$$

not


 * $$V_\alpha := \bigcup_{\beta \in \alpha} \mathcal{P} (V_\beta)$$

So, this was not even an assumption made in the rest of the article. I don't know that I've ever met any "ordinary mathematicians" but I can only guess they would just use a different representation... --192.75.48.8 (talk) 22:06, 11 March 2015 (UTC)


 * I see that I was reverted but without response. Does the person re-instating the claim think that I am mistaken for thinking ω+ω can be represented? Or does he or she think we "ordinarily" care how objects are represented? Or what? --192.75.48.8 (talk) 14:00, 12 March 2015 (UTC)


 * Since the the statement about "ordinary mathematics" hinges on how "ordinary mathematics" is defined, the qualifying statement is warranted as was explained in the edit summary. For ordinals, $α ∈ β$ if and only if $α < β$ provided that I recall this correctly. Further, I don't think ordinal arithmetic depends on how ordinals are defined. Moreover, their membership in some $V_{α}$ seems to be definition-independent. Maybe ordinal arithmetic is not ordinary mathematics, but then you need to explain that statement. You will invariably then get back to something alike what was in the article before your edits. I have no strong feelings about this. YohanN7 (talk) 14:31, 12 March 2015 (UTC)
 * OK, so 192 is correct on the immediate point of dispute. You can easily represent small ordinals &mdash; certainly, all countable ones, and indeed quite a bit more &mdash; within V&omega;+&omega;, and define arithmetic on them.  It's only if you want them to be von Neumann ordinals that you need to go higher in rank.
 * That said, the "ordinary mathematics" language is still problematic. There are plenty of propositions that are arguably part of ordinary mathematics (say, "every projection of a co-analytic set of reals is Lebesgue measurable") that are not naturally justified by considerations at the level of V&omega;+&omega;. --Trovatore (talk) 15:41, 12 March 2015 (UTC)
 * Ok, I stand corrected. But how are countable ordinals represented within V&omega;+&omega;, and how is that representation "natural" in any sense? YohanN7 (talk) 16:28, 12 March 2015 (UTC)
 * Well, a countably infinite ordinal is just the order-type of a wellordering of the naturals. Wellorderings of the naturals can be represented as sets of ordered pairs of naturals.  That doesn't give you a canonical representative, because there are different wellorderings with the same order type, but you can take equivalence classes at the price of moving up one rank. --Trovatore (talk) 17:01, 12 March 2015 (UTC)
 * Well, I was only objecting to the particular point, but since you mention it: I think the article's mention of "axiom of replacement" is confused as well. I personally would distinguish between "what is provable in Zermelo set theory" and "what is true in V&omega;+&omega;" (as proven by ZFC, or whatever else). What axioms we use to prove things about sets of reals is really not relevant to the question of whether "ordinary mathematics" takes place there or not. For that matter, such axioms may have implications for V&omega;, but I don't think anybody says V&omega; is inadequate as a model of finitary mathematics. --GodMadeTheIntegers (talk) 17:21, 12 March 2015 (UTC)

Condensed definition
One reads:

An equivalent definition sets
 * $$V_\alpha := \bigcup_{\beta < \alpha} \mathcal{P} (V_\beta)$$

for each ordinal α, where $$\mathcal{P} (X) \!$$ is the powerset of $$X$$.

Isn't that true only for limit ordinals α ? Should we correct it? --Agatino Catarella (talk) 09:43, 13 August 2023 (UTC)
 * The definition works for successor ordinals as well. The $$\bigcup$$ part turns out to be superfluous, as all the sets from previous stages are also contained in the last stage in the union.
 * I don't personally like this sort of almost "trickological" terseness, but there seem to be people who think that it is not trickological and in fact is informative. We could probably argue about that for days.  If this formulation is in high-quality sources it probably makes sense to include it; otherwise not. --Trovatore (talk) 18:10, 13 August 2023 (UTC)


 * If you use the condensed definition, it becomes easier to prove that the sequence of sets has the desired properties. For example, showing that the sequence is nested follows immediately from big-cup. JRSpriggs (talk) 00:19, 14 August 2023 (UTC)


 * I mean, sure, if you just want to prove that particular fact, maybe you save five seconds. If you actually want to understand what's going on, it seems to me, you still need to understand that the limit and successor cases are different.  In the limit case, nothing new is added that wasn't there before; in the successor case, it is.
 * Put another way, the standard definition takes powersets at successors and unions at limits. The definition under discussion says, ha ha, I can just do both at every step.  At the successor step, the union is redundant, whereas at the limit step, the powersets don't accomplish anything, but I can do it.
 * Anyway that's my aesthetic reaction. If this is done in a high-quality source (Kunen, Jech, Halmos, Enderton, that category) I would withdraw my objection.  If it's present in a lower-quality source, well, we should look at it and discuss. --Trovatore (talk) 16:25, 14 August 2023 (UTC)


 * Thanks a lot everyone for the explanations. I don't like aither too much those definitions that rely on side effects, but that's probably a matter of tastes, indeed. Maybe we could add a word to explain the disappearance of the "limit" word, as I am probably not the first one to wonder. Not sure. --Agatino Catarella (talk) 04:20, 15 August 2023 (UTC)
 * Note that there's a tangentially related discussion at talk:ordinal arithmetic. --Trovatore (talk) 20:37, 15 August 2023 (UTC)
 * This kind of condensed definition seems to be favored by Cohen, at least in the book "Set Theory and the Continuum Hypothesis". The particular definition is in Section II.5 "The Axiom of Regularity", where he first defines the concept of rank in terms of rank functions, and then notes "for future reference that for each ordinal α, we can prove the existence of the set of sets of rank α. Namely, by transfinite induction define $$S(\alpha) = \mathcal{P}(\bigcup_{\beta < \alpha} S(\beta))$$". (Note that the power set operation is on the outside, because "the set of sets of rank ≤α" should be the power set of $$V_\alpha$$.) In Chapter III he defines the constructible universe similarly as "$$M_0 = \varnothing$$ and $$M_\alpha = (\bigcup_{\beta < \alpha} M_\beta)'$$" (where $$x'$$ is his notation for $$\mathrm{Def}(x)$$). Bbbbbbbbba (talk) 06:41, 28 May 2024 (UTC)