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Many of us presume that mathematics studies objects. In agreement with this, set theorists often say that they study the well founded hereditarily extensional objects generated ex nihilo by the "process" of repeatedly forming the powerset of what has already been generated and, when appropriate, forming the union of what preceded.

But the practice of set theorists belies this, since they tend—for instance, in the theories of inner models and large cardinal embeddings—to study classes that, on pain of contradicting the standard axioms, are never "generated" at any stage of this process. In particular, faced with the independence results, many set theorists suggest that each statement about sets—regardless of whether it be independent of the standard axioms, or indeed of whether it be formalizable in the first order language of set theory—is either true or false about the class $V$ of all objects formed by the above mentioned process. For them, set theory is the attempt to uncover the truth about $V$.

This tendency is at odds with what I said set theorists study, because proper classes, though well founded and hereditarily extensional, are not objects. I do not mean just that proper classes are not sets.

Rather, I suggest that no tenable distinction has been, nor can be, made between well founded hereditarily extensional objects that are sets, and those that aren't.

Of course, this philosophical claim cannot be proved.

Instead, I offer a persuasion that I hope will provoke you to enlighten me with your thoughts.

Suppose the distinction were made. Then in particular, $V$ is an object but not a set. Prima facie, it makes sense to speak of the powerclass of $V$—that is, the collection of all hereditarily well founded objects that can be formed as "combinations" of objects in $V$. This specification should raise no more suspicions than the standard description of the powerset operation; the burden is on him who wishes to say otherwise.

With the powerclass of $V$ in hand, we may consider the collection of all hereditarily well founded objects included in it, and so on, imitating the process that formed $V$ itself. Let $W$ be the "hyperclass" of all well founded hereditarily extensional objects formed by this new process. Since we can distinguish between well founded hereditarily extensional objects that are sets and those that aren't, we should be able to mirror the distinction here, putting on the one hand the proper hyperclasses and on the other the sets and classes.

Continuing in this fashion, distinguish between sets, classes, hyperclasses, $n$-hyperclasses, $\alpha$-hyperclasses, $\Omega$-hyperclasses, and so on for as long as you can draw indices from the ordinals, hyperordinals, and other transitive hereditarily extensional objects well ordered by membership, hypermembership, or whatever. It seems that this process will continue without end: we will never reach a stage where it does not make sense to form the collection of all well founded hereditarily extensional objects whose extensions have already been generated. We will never obtain an object consisting in everything that can be formed in this fashion.

For me, this undermines the supposed distinction between well founded hereditarily extensional objects that are sets, and those that aren't. Having assumed the distinction made, we were led to the conclusion of the preceding paragraph. But that is no better than the conclusion that proper classes, including $V$ itself, are not objects. Indeed, it is worse, for in arriving at it we relegated set theory to the study of just the first two strata of a much richer universe. Would it not have been better to admit at the outset that proper classes are not objects? If we did that, would set theory suffer? In particular, how would it affect the idea that each statement about sets is either true or false?

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While similar questions have already been posed on MO, I thought it worthwhile to approach the topic from this angle. I apologize if this upsets anyone. –  Cole Leahy Jul 31 '11 at 22:45
    
Although I've not thought about these things for some decades, it seems to me that this is an unusually-well-considered question! :) –  paul garrett Jul 31 '11 at 22:47
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Can you say what you take the word "object" to mean? –  Tom Leinster Aug 1 '11 at 5:15
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I'm surprised it hasn't been closed as "not a real question" yet (though I will not vote to close because I'm interested in the discussion). –  Qfwfq Aug 1 '11 at 10:39
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Voted to close. MO is not a discussion site. –  Gerald Edgar Aug 2 '11 at 14:18

4 Answers 4

up vote 21 down vote accepted

Proper classes are not objects. They do not exist. Talking about them is a convenient abbreviation for certain statements about sets. (For example, $V=L$ abbreviates "all sets are constructible.") If proper classes were objects, they should be included among the sets, and the cumulative hierarchy should, as was pointed out in the question, continue much farther, but in fact, it already continues arbitrarily far.

In particular, when I talk about statements being true in $V$, I mean simply that the quantified variables are to be interpreted as ranging over arbitrary sets. It is an unfortunate by-product of the set-theoretic formalization of semantics that many people believe that, in order to talk about variables ranging over arbitrary sets (or arbitrary widgets or whatever), we need an object, a set, that contains all the sets (or all the widgets or whatever). In fact, there is no such need unless we want to formalize this notion of truth within set theory. Anyone who wants to formalize within set theory the notion of "truth in $V$" is out of luck anyway, by Tarski's theorem on undefinability of truth.

Considerations like these are what prompt me to view ZFC as a reasonable foundational system, in contrast to Morse-Kelley set theory.

A detailed explanation of how to use proper classes as abbreviations and how to unabbreviate statements involving them is given in an early chapter of Jensen's "Modelle der Mengenlehre". (The idea goes back at least to Quine, who used it not only for proper classes but even for sets, developing a way to understand talk about sets as being about "virtual sets" and avoiding any ontological commitment to sets.)

Finally, I should emphasize, just in case it's not obvious, that what I have written here is my (current) philosophical opinion, not by any stretch of the imagination mathematical fact.

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"...many people believe that, in order to talk about variables ranging over arbitrary sets ... we need an object, a set, that contains all the sets ... " But this can be true by definition, since we can just re-ask the original question with the predicate "is an object" replaced by "can be quantified over". Then the conclusion is that we cannot quantify over V. That's a problem! I guess we should explain why this substitution cannot be done, or how it causes the reasoning in the question to fail where it succeeded before, or how it was not sound to begin with. –  Daniel Mehkeri Aug 1 '11 at 8:19
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This is a very reasonable viewpoint, which I think is shared by many set theorists. But I think the deeper philosophical question, which I am not in a position to answer, is what a mathematical object is in the first place, and whether the act of discussing something in mathematics makes it into an object. –  Carl Mummert Aug 1 '11 at 11:12
    
@Daniel: quantification is specified by inference rules. That is, at the basic level quantification is a logical operation (or linguistic if you will), which does not come with a semantic pre-attached to it. If we speak about the meaning of quantification, that is a different matter. But here again we see that no collections or classes are actually needed because there are many kinds of semantics, some of which are not of the Tarskian kind. And it is enough to have one such semantics to show that quantification does not presuppose existence of classes or collections. –  Andrej Bauer Aug 2 '11 at 5:33
    
@Andrej: Andreas even did mention this at the end of his answer. But under non-Tarskian semantics, we don't have to presuppose "existence" of sets either (he mentioned this too). It sort of sounds like, whatever "existence" means here, it's not what makes sets different from proper classes. I guess at least it requires further explanation. –  Daniel Mehkeri Aug 3 '11 at 3:30
    
FWIW (and it will likely be worth less to mathematicians than historians of logic and philosophy) the idea you attribute to Quine sounds more like Russell and Whitehead, who in PM *20.01 and *20.02 show how to eliminate sets for propositional functions. The status of propositional functions, however, remains a hotly debated topic among Russell scholars with recent work by Gregory Landini (my dissertation adviser) revealing that Russell likely had in mind a substitutional semantics for quantification over propositional functions. Landini's interpretation is challenged by Bernard Linsky. –  Jeremy Shipley Aug 4 '11 at 3:03

(Insert standard and obvious disclaimers about opinion vs. fact.)

Of course proper classes are mathematical objects. The fact that we can say things like "the proper class M is a model of set theory" means that proper classes must be objects, if as you say "objects are the bearers of properties" in mathematics.

What proper classes are not, and the only thing they are not, is elements of a model of ZF. They are elements of a model of NBG, which is just as good a first-order theory, and foundation of mathematics, as ZF is. And they are elements of the ambient theory in which we speak about models of ZF. (To be more precise, there are certain elements in a model of NBG called "proper classes" which correspond, in a certain precise way, to certain elements of the ambient theory of a model of ZF, which justifies their being given the same name.)

Regarding the latter point: all the time set-theorists prove theorems with hypotheses like "for any proper-class model of ZF, ..." or state the axiom of replacement as "for any class-function whose domain is a set, its range is also a set". Of course such theorems and axioms cannot be formalized within ZF as single theorems, but they are stated in set theory books as single sentences (as they must be, since the set theory books have finite length)—and as such they are single theorems of mathematics about objects called "proper classes", despite their not being single theorems of ZF.

I'm guessing the insistence that proper classes are "not objects" stems from a belief that all of mathematics takes place in ZF. I find such a claim much harder to justify than the slightly different claim that all of mathematics could be coded into ZF, were anyone so inclined. It could also, of course, be coded into many different foundational systems.

As to the question of how to distinguish between "well founded hereditarily extensional objects that are sets and those that aren't," I would say that a well-founded hereditarily extensional object that is an element of some model of ZF is a "set" according to that model, whereas one which isn't, isn't. Obviously this varies with the model we choose (and only makes sense in some ambient context where we have "models" to speak about), but that's the way things are. There is no Platonic universe of "real" mathematical objects that contains "all" the well-founded hereditarily extensional things, from which we could ask to separate out those that "are" sets from those that "aren't".

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This will sound like splitting hairs, but in a technical sense we cannot say that "the class of all sets is a model of ZF". This is because the model relation can only be defined for structures that are sets. Otherwise we run into problems with Tarski's undefinability of truth. We do, however, usually assume that all axioms of ZF hold in the the class of all sets, but that is a metamathematical statement. –  Stefan Geschke Aug 2 '11 at 5:49
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@Stefan: We can say it just fine, in a perfectly technical and precise sense. We just can't formalize it as a single statement inside the model of ZF in question. But "the class of all sets in this model of ZF is a model of ZF" is a perfectly true (though somewhat tautological) statement about any model of ZF, in any context in which one can talk about a model of ZF. –  Mike Shulman Aug 3 '11 at 3:56

I think Andreas provided an excellent answer, and he pointed out that his answer expresses his philosophical opinion, not some absolute mathematical truth. I wanted to add some things, though.

Russell has shown to us that unrestricted freedom to form "objects" leads to a contradiction. What you have in mind seems to be some sort of type theory, as your "hyperclasses" have indices ("$\alpha$-hyperclasses") and only consist of objects of lower index. This should avoid the Russell paradox. However, I always thought that it is one of the great achievements of theories like ZFC that they got rid of the technical complications of type theory.

Of course, there are set theories where proper classes exist as objects, for example Neumann-Gödel-Bernays set theory (NGB). NGB has the same consistency strength as ZFC, so in some sense it doesn't give anything new.

Finally, mathematics seems to work out just fine in ZFC. Even though proper classes are not objects, this doesn't seem to be a serious obstruction and we can do everything that is normally done in mathematics.

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There are some things, though, that people "normally" do in some areas of mathematics and that are at least harder to do in ZFC because of the lack of classes (I'm thinking in things like large categories, or categories whose objects are large categories themselves). –  David FernandezBreton Apr 14 '12 at 4:40

Let me offer another answer in counterpoint to Andreas's answer, by pointing out a number of cases in set theory where it seems that a second-order treatment of classes, as in Goedel-Bernays set theory, seems fruitful in contrast to the definable-classes-only approach.

  • First, much of our understanding of large cardinals is based upon a felicitous use of large cardinal embeddings $j:V\to M$, which are all proper classes. And set-theorists routinely quantify over the meta-class collection of such embeddings. For example, a cardinal is measurable if it is the critical point of such an embedding; it is strong if for any $\theta$ there is such an embedding with $V_\theta\subset V$; it is supercompact if such embeddings can be found with $M^\theta\subset M$. Although in each case we have a first-order combinatorial equivalent to the large cardinal concept in terms of the existence of certain kinds of measures or extenders on certain sets, nevertheless it is the embedding characterizations that have a robust and strongly coherent power that unifies our understanding of the large cardinal concepts. This seems to be a case in which treating the embeddings as objects has deepened our knowledge.

  • The Kunen inconsistency result, the assertion that there is no nontrivial elementary embedding $j:V\to V$, becomes trivial when one treats all classes as definable. One can easily rule out all such definable $j$, if one only cares to consider the case in which $j$ is first-order definable with parameters, and one needs neither the axiom of choice nor any infinite combinatorics to do it. (Just argue like this: the question of whether a given formula $\varphi(x,y,p)$ with parameter parameter $p$ defines such a $j$ is a first-order property of $p$, and so one can define $\kappa$ to be the least possible critical point of such a $j$ arising from any $p$, and this contradicts the fact that $\kappa\lt j(\kappa)$, since $j(\kappa)$ would also be defined this way.) The various formalizations of the Kunen inconsistency is explained in the first part of our recent paper on Generalizations of the Kunen Inconsistency. Note that Kunen formalized his theorem in Kelly-Morse set theory, in order to have a way of expressing the elementarity of $j$, but it turns out to be possible to formalize this in GBC.

  • Class forcing is vital to much of our understanding of the relative consistency of global assertions, such as the full class version of Easton's theorem or the fact that supercompact cardinals are relatively consistent with GCH and with V=HOD. But a development of class forcing is most direct in a context, such as Goedel-Bernays set theory, where classes can be treated as objects. For example, when forcing the GCH with class forcing, the generic class will not be definable in the forcing extension, so this is a case where in order to achieve the extension we most naturally want to consider non-definable classes.

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For many large cardinal properties, if a cardinal has the property, then a certain kind of set witnesses it; for instance, $\kappa$ is measurable iff there is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$. But for various "super" properties, it seems like only a proper class witnesses; for instance, $\kappa$ is supercompact iff for all $\lambda > \kappa$ there is a $\kappa$-complete normal fine ultrafilter on $P_{\kappa}(\lambda)$, so we need a proper class of ultrafilters. (Continued below.) –  Cole Leahy Aug 9 '11 at 19:00
    
Now this is no barrier to our writing a first-order formula satisfied, relative to a model, by all and only the supercompact cardinals of the model. But, assuming that a given set $\kappa$ either is or isn't "in the real world" a supercompact cardinal, is a special philosophical difficulty raised by the fact that the supercompactness of $\kappa$ is never settled at any level of the cumulative hierarchy? Or is this phenomenon not particular to large cardinals? –  Cole Leahy Aug 9 '11 at 19:00
    
Cole, all the standard large cardinal properties, including supercompact cardinals, are first-order expressible, and their corresponding embeddings $j:V\to M$ are all uniformly definable. So there is no problem ultimately with treating them in line with Andreas's philosophy. That is, one doesn't need classes as objecs to treat supercompact and other large cardinals. My point, instead, was that by treating the embeddings as objects, one can ignore the irrelevant complexities added by considering those definitions, and just work with the embeddings $j$ naturally as second order objects. –  Joel David Hamkins Aug 9 '11 at 19:27
    
And furthermore, this manner of considering the large cardinal embeddings has been fruitful. –  Joel David Hamkins Aug 9 '11 at 19:28
    
Joel, we may be talking past each other. I recognize, and even noted above, that supercompactness is first-order expressible. However, I wanted to flag a way that supercompactness seems to differ from many other properties a set could have. Namely, supercompactness is a property that a cardinal possesses not in virtue of its relation to set-many other sets, but rather to class-many other sets. No matter whether supercompactness is cashed out in terms of embeddings, extenders, or ultrafilters, that $\kappa$ is supercompact is not a fact about the relation of $\kappa$ to any $V_{\alpha}$. –  Cole Leahy Aug 10 '11 at 3:29

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