When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Question

Every once in a blue moon it actually matters that some mathematical entity which might a priori only be a class is in fact a set. For clarification, here are some examples of what I do not mean:

A) Some colleagues of mine once made the following disclaimer: 'The "set" of stable curves does not exist, but we leave this set theoretic difficulty to the reader.' These colleagues (names withheld to protect the innocent) are of course fully aware of the fact that strictly speaking, the class of all stable curves, or topological spaces, or groups, or any the other usual suspects customarily formalized in terms of structured sets, cannot itself be a set. While they recognize that the structure they study is transportable along arbitrary bijections between members of a proper class of equinumerous sets, they also recognize that in their setting this same transportability could justify a technically sufficient a priori restriction to some fixed but otherwise arbitrary underlying set: that is, the relevant large category has a small skeletal subcategory. (Exercise: precisely what makes this work in the example given?) In such cases, the set versus class pecadillo is an essentially victimless one, perhaps barring discussions of the admissibility of Choice, expecially Global Choice.

B) There are various contexts in which seemingly unavoidable size issues are managed through the device of Grothendieck Universes. Such a move beyond ZFC might be regarded as cheating, sweeping the issue under the carpet for all the right reasons. Allegations of this nature regarding the use of derived functor cohomology in number theory, as in the proof of Fermat's Last Theorem, can now be laid to rest, as Colin McLarty has nicely shown in "A finite order arithmetic foundation for cohomology" http://arxiv.org/abs/1102.1773.

C) Set theory itself is replete with situations where the set versus class distinction is of paramount importance. For just one example, my very limited understanding is that forcing over a proper class of conditions is not for the unwary. I'd be interested to hear some expert elucidation of that, but my question here is in a different spirit.

With these nonexamples out of the way, I have a very short list of examples that do meet my criteria.

1) Freyd's theorem on the nonconcretizability of the homotopy category in "Homotopy is not concrete" http://www.tac.mta.ca/tac/reprints/articles/6/tr6abs.html. By definition, a concretization of a category is a faithful functor to the category of sets. The homotopy category (of based topological spaces) admits no such functor. The crux of the argument is that while any object of a concretizable category has only a set's worth of generalized normal subobjects, there are objects in the homotopy category - for example $S^2$ - which do not have this property (page 9). The original closing remark (page 6) mentions another nonconcretizability result, for the category of small categories and natural equivalence classes of functors. A purist might try to disqualify the latter as too `metamathematical', but the homotopy example seems unassailable.

2) A category in which all (co)limits exist is said to be (co)complete; a bicomplete category is one which is both complete and cocomplete. Freyd's General Adjoint Functor Theorem gives necessary and sufficient conditions for the existence of adjoints to a functor $\Phi:{\mathfrak A}\rightarrow{\mathfrak B}$ with $\mathfrak A$ (co)complete. Let us say that a functor which preserves all limits is continuous, and that one which preserves all colimits is cocontinuous. A bicontinuous functor is one which is both continuous and cocontinuous.

Let us say that $\Phi$ is locally bounded if for every $B\in {\rm Ob}\,{\mathfrak B}$ there exists a set $\Sigma$ such that for every $A\in{\rm Ob} \,{\mathfrak A}$ and $b\in{\rm Hom}_{\mathfrak B}(B,\Phi A)$ there exist $\hat{A}\in{\rm Ob}\,{\mathfrak A}$ and $\hat{b}\in{\rm Hom}_{\mathfrak B}(B,\Phi\hat{A})\cap\Sigma$ such that
$b=(\Phi \alpha)\hat{b}$ for some $\alpha\in{\rm Hom}$ $_{\mathfrak A}$ $(\hat{A},A)$, and that $\Phi$ is locally cobounded if for every $B\in {\rm Ob}\,{\mathfrak B}$ there exists a set $\Sigma$ such that for every $A\in{\rm Ob}\,{\mathfrak A}$ and $b\in{\rm Hom}_{\mathfrak B}(\Phi A,B)$ there exist $\hat{A}\in{\rm Ob}\,{\mathfrak A}$ and $\hat{b}\in{\rm Hom}_{\mathfrak B}(\Phi\hat{A},B)\cap\Sigma$ such that $b=\hat{b}(\Phi \alpha)$ for some $\alpha\in{\rm Hom}_{\mathfrak A}(A,\hat{A})$. In the literature these are known as the Solution Set Conditions.

Theorem. Let $\Phi:{\mathfrak A}\rightarrow{\mathfrak B}$ be a functor, where $\mathfrak B$ is locally small.

$\star$ If $\mathfrak A$ is complete then $ \Phi$ admits a left adjoint if and only if $\Phi$ is continuous and locally bounded.
$\star$ If $\mathfrak A$ is cocomplete then $ \Phi$ admits a right adjoint if and only if $\Phi$ is cocontinuous and locally cobounded.

See pages 120-123 of MacLane's "Categories for the working mathematician".

The local (co)boundedness condition has actual content. For example:

a) The forgetful functor ${\bf CompleteBooleanAlgebra}\rightarrow{\bf Set}$ is continuous but admits no left adjoint.

b) Functors ${\bf Group}\rightarrow {\bf Set}$, continuous but admitting no left adjoint, may be obtained as follows: let $\Gamma_\alpha$ be a simple group of cardinality $\aleph_\alpha$ (e.g. the alternating group on a set of that cardinality, or the projective special linear group on a 2-dimensional vector space over a field of that cardinality) and take the product (suitably construed), over the proper class of all ordinals, of the functors ${\rm Hom}_{\bf Group}(\Gamma_\alpha,-)$.

c) Freyd proposed another interesting example (see page -15 of the Foreword to "Abelian categories" http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html) of a locally small bicomplete category $\mathfrak S$ and a bicontinuous functor $\Phi:{\mathfrak S}\rightarrow {\bf Set}$ which admits neither adjoint: loosely speaking, the category of sets equipped with free group actions, and the evident underlying set functor.

Does anyone know of any other examples, especially fundamentally different examples?

Finally, one could focus critical attention on the very question posed. To what extent does the strength and flavor of the background set theory matter? Force of habit and comfort have me implicitly working in some material set theory such as ZF, perhaps a bit more if I want to take advantage of Choice, perhaps a bit less if I prefer to eschew Replacement. Indeed, I have actually checked that example b) may be formulated in the absence of Replacement: while the von Neumann ordinals are no longer available, the same trick already used to give a kosher workaround to the illegitimate product over all ordinals further shows that an appropriate system of local ordinals suffices for the task. I am also quite interested in hearing what proponents of structural set theory have to say.

Answer 1

In my experience in analysis, basically the only place where it is actually important to distinguish sets from proper classes arises when one wishes to invoke Zorn's lemma to locate a maximal object in some non-empty partially ordered set $X$ in which all chains are bounded (e.g. to create a maximal proper subspace, a maximal filter, a maximally defined bounded linear functional, etc.). Here it is crucial that $X$ is "small" enough to be an actual set (e.g. it is a collection of subsets of some space $V$ that is already known to be a set, or a collection of functions from $V$ to yet another set). For instance, one cannot use Zorn's lemma to construct a maximal set in the class of all sets, or a maximal group in the class of all groups, or a maximal vector space in the class of all vector spaces, despite the fact that in each of these classes, any chain has an upper bound (the direct limit). (Such maximal objects, if they existed, would soon lead to contradictions of the flavour of Russell's paradox or the Burali-Forti paradox; not coincidentally, one of the standard proofs of Zorn's lemma proceeds by contradiction, using the axiom of choice to embed all the ordinals into $X$, which can then be used to set up the Burali-Forti paradox.)

To put it another way: regardless of one's choice of foundations, it is clearly mathematically desirable to be able to easily locate maximal objects of various types; but it is obviously also desirable for the existence of such maximal objects to not lead (or mislead) one into paradoxes of Russell or Burali-Forti type. ZFC, with Zorn's lemma on one hand and the set/class distinction on the other, manages to achieve both of these objectives simultaneously. Presumably, many other choices of foundations (particularly those which are essentially equivalent to ZFC in a logical sense) can also achieve both objectives at once, but I usually don't see these points emphasised when such alternative foundations are presented in the literature.

Answer 2

Your question does not seemed aimed at set theorists, but let me give a set theorist's answer.

I view the set/class distinction as analogous to and ultimately no more problematic really than the other distinctions of size that are commonly made in mathematics.

For example, we study the finite groups as a robust, coherent collection, and we are untroubled by the fact that there are many than finitely many isomorphism types. We just don't find it confusing that there are infinitely many finite groups. (For example, we don't expect to deduce by Zorn's lemma that there are maximal finite groups.) Or we study the collection of countable graphs, while realizing that there are uncountably many instances even on the same set of vertices. More generally, we might look at $\kappa$-dense topological spaces, or at all structures of a given type of size less than a cardinal $\kappa$, or at spaces of a given dimension or rank, and so on.

These distinctions of size are extremely common and part of the way that we think mathematically; these distinctions are part of the way that we carve up our mathematical universe at its joints. Similarly, we may handle the set/class distinction, which is of the same character, neither especially mysterious or problematic.

In each case, we have to pay attention to the details of the mathematical constructions that we employ, in order that these constructions not take us out of the class in focus.

As you say, set theory is replete with these considerations of size and similar distinctions. The entire large cardinal hierarchy is an investigation of different sizes of infinity. The Grothendieck universe concept, arising at the entryway of that hierarchy, is a such measure of size distinction, usually considered a bit crude or clumsy by set theorists, but useful for non-set-theorists because it is easy to understand. Meanwhile, set theory is full of other subtler universe concepts: the levels of the arithmetical and projective hierarchies provide "universes" of complexity for countable objects; the various cut-off universes $H_\kappa$, $L_\kappa$, $V_\kappa$ are often used as local universe concepts; the proper-class sized inner models $L$, $\text{HOD}$, $L(\mathbb{R})$, $L[0^\sharp]$ and so on provide limitations of the background universe that is not just of "size", but of set-theoretic complexity. In broad strokes, all these limitations affect mathematical argument in a similar way, since one must pay attention to which kinds of constructions might take you beyond the limitation that has been set.

The set/class distinction is just one more such distinction.

Answer 3

Terry Tao has already mentioned Zorn's Lemma in order to find maximal elements in small partial orders. More generally, colimits in categories usually only exist for small index categories. In fact, every category admitting colimits for all index categories is equivalent to a partial order.

Another typical example of this kind is the small object argument. It says that any set of morphisms in a category with certain conditions produces a functorial weak factorization system. The transfinite construction doesn't stop when we start with a class of morphisms.

Another example: A cocomplete symmetric monoidal category is closed if and only if all functors $X \otimes -$ satisfy the solution condition. Todd Trimble has given an example where this fails. It is interesting that being closed is only a property of the data, but the property seems to depend on the size.

Answer 4

Proper classes come up when you exhaust the means of forming sets. You need a set when you need to know the means of set theory have not been exhausted -- for example when you want to go on and form a colimit of the structures you have formed so far. Exactly when the means are exhausted, depends on what means of forming sets you have.

First take an example that exhausts second order arithmetic but does not exhaust Zermelo set theory (or simple type theory): the etale fundamental group of an arithmetic scheme. There is no universal cover like the ones for topological spaces and this is not a logical or set theoretic problem but inherent in the situation. (The scheme has etale covers of any finite degree, so a universal cover could have no finite degree.) So Grothendieck and others formed the colimit of all symmetries of the (non-universal, actually existing) etale covers. Second order arithmetic suffices to give the symmetry group of any one etale cover, but because we want the colimit of all these, we need an uncountable group. Second order arithmetic will not produce that. Third order will.

Grothendieck and Dieudonne often found they wanted colimits sort of like this, over all cases of some structure, but not just all that exist in second order arithmetic. Naively put, they wanted all that exist in set theory. Maybe all algebras over some ring, or all finitely generated algebras. They knew there is a big difference between those examples, since there is not even a set of all algebras over a ring up to isomorphism (in any set theory they considered). Choosing one countably infinite set of generators will give you a set of all finitely generated algebras over that ring up to isomorphism. But in either case they did not want to bother with such details. And they were all the more eager to avoid analogous details in more complex cases.

If you really want to talk about all sets, or all natural weak equivalences of functors from Top to Top, or all generalized normal subobjects of $S^2$ in the homotopy category then you are exhausting the means of set theory (though the last two cases are less obvious than the first).

Grothendieck and Dieudonne appreciated the point perfectly. They knew workarounds to fit some of their larger constructions into ordinary set theory, and they were confident other workarounds could be found. But they were not interested in that. They saw that when they used all sets etc., it was not "all" in any metaphysical sense. It was all those constructed by the ordinary means of set theory, so they posited one non-ordinary means of constructing sets: each set is contained in a universe. At any point they work inside some universe, so what would be proper classes in ordinary set theoretic accounts are sets in the next larger universe.