Classifying whether a collection of objects is a set

I'm wondering whether or not there is a definite criterion for determining whether a collection of objects is a set. In the literature, I've recently encountered some obscure examples (in Riemann Surface Theory, to be precise) of naturally-arising collections of objects which are not sets.

Here is some pseudo-rigor to make the question more precise:

Let $S$ be a collection of objects. Is there some property $P$ so that $S$ is a set iff $S$ has property $P$?

Solution 1:

In ZF-style foundations, the name of the game is rank (which annoyingly doesn't have its own wikipedia page). If you have defined a class $C$ of sets representing mathematical objects in ZF (or ZFC, or NBG or what-have-you) then it is a set if and only if there is an ordinal $\alpha$ such that every element of $C$ has rank less than $\alpha.$

In practice, this is usually decidable, but it doesn't need to be since we can easily cook up a stupid example to the contrary: let $C$ be defined as the class where for any set $x,$ $x\in C$ if and only if the continuum hypothesis holds. So $C$ is a proper class if and only if CH holds, and so assuming we aren't using axioms strong enough to decide CH, it is undecidable whether $C$ is a set.

(That was the "useful" answer. A less useful answer is that a class is a set iff it is co-extensive with a set. We could write this in the first order language of set theory as: $\exists y \forall x(x\in C\leftrightarrow x\in y)$ where here $x\in C$ is shorthand for the first-order formula $\varphi_C(x)$ that defines membership in the class. It just so happens that in ZF, this is provably equivalent to the more satisfying property above.)

Solution 2:

In ZF and related theories, as indicated in another answer, the key to whether a class is a set is whether it has a rank, meaning that it appears at some point in the process of forming sets by starting with the empty set and iterating (throughout the ordinals) the operations of taking power sets and collecting what you have so far.

The drawback of the above is that some working knowledge of ordinals is needed even to understand the statement. However, as a corollary, there is a different criterion that is also useful in practice. To state it, we need the notion of transitive closure. The thing to keep in mind is that in ZF every actual object is a set. Given a class $x_0=x$, its elements are sets and so we can collect them together by taking the union of $x$, $x_1=\bigcup x_0$. The elements of the elements of $x$ are also sets, so we can collect them together by considering $x_2=\bigcup x_1$, etc. Let $x_\infty=\bigcup_{n\in\mathbb N}x_n$. This is the transitive closure of $x_0$.

Ok, the criterion:

A class $X$ is a set if and only if there is a bound on the size of the members of $X_\infty$.

This is useful in practice: The class of all countable groups, for instance, is a proper class. It is true that all such groups are bounded in size, but their elements don't need to be: Given any group $G$, you can pick a monstruously large set $M$ and obtain a group isomorphic to $G$ simply by replacing the identity of $G$ with $M$.

Many natural classes appearing in practice are closed under isomorphism, and a similar silly construction as in the example above shows that they are proper classes.

(The universe of all sets can be stratified according to rank, for any ordinal $\alpha$, the set of all sets of rank smaller than $\alpha$ is denoted $V_\alpha$, and any set belongs to some $V_\alpha$.

Sets can also be stratified according to the size of their transitive closure. For any cardinal $\kappa$, the sets $x$ whose transitive closure has size strictly smaller than $\kappa$ is denoted $H(\kappa)$ or $H_\kappa$, and any set belongs to some $H_\kappa$. There is a small technicality that doesn't come into play in the above; see here.)

Solution 3:

Traditionally speaking, along the standard line of set theory, i.e. ZFC, "set" is not definable, the axioms of ZFC function to lay down characteristics about sets, so every object in the domain of discourse of ZFC is a set. If we allow classes on top of ZFC, like the case of NBG or MK, then here you can speak of sets as special case of classes, and in these theories sets can be defined as classes that are elements of classes. However, this capability of being an element of a class is connected to a concept related to size comparisons between classes, i.e. their cardinality. One version of NBG\MK depicts sets as classes that are strictly smaller than the class of all elements, i.e. the universe, in which case we'll have global choice. You can also use the Hierarchy notion to define sets in a class theory as sets being classes that are sub-classes of a stage in the cumulative Hierarchy [defined by Von Neumann]. Of course this emphasizes the well founded structure of sets as an additional feature next to them being collectible as elements of a classes, as well as transferring to them the size notions of the relevant class theory related to enabling collections of being elements.