Cardinality restrictions: Too many to be included in a set While reading literature sometimes I stumble on such passages as


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*In the first-order case, we cannot require M to contain a distinct world for each model of the first-order non-modal language, for there are too many models to form a set.

*Since there are too many ordinals to form a set, there are too many propositions to form a set.

*By hypothesis, such a sentence can contain too many variables to form a set.
What does it mean "too many to form a set"? What justifies such cardinality restrictions?
 A: Tl;dr: it's not "a priori," but rather a feature of the particular choice of set theory we make.

As you know, there are various paradoxes in "naive" set theory which basically tell us that in any consistent theory of sets, certain collections can't actually be sets. The two most important examples are probably the following:

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*Russell's paradox rules out the sethood of the collection of all sets not containing themselves.


*The Burali-Forti paradox rules out the sethood of the collection of all ordinals (under any appropriate definition of "ordinal").
Rigorous set theory begins with the task of whipping up an axiomatic framework (or more than one!) which avoids these paradoxes. However, different theories may avoid paradox in different ways. For example, in positive set theories the collection of sets which do contain themselves is a set! It's not at all clear that this set is "bigger" in any sense than the Russell collection. The identification of size as the key criterion for sethood is a feature of some, but not all, set theories.

... But in particular, it is a feature of $\mathsf{ZFC}$, which is the generally-accepted default set theory, and its many relatives. There, the relevant fact is the following:

For any set $x$, there is some set $y$ such that there is no surjection from $x$ to $y$.

(In fact, we can take $y$ to be an ordinal!) This means that we can show that a given class $C$ is not a set by showing that it is "too big" in the sense that we can surject $C$ onto any arbitrary set.
One way to prove this is via the Burali-Forti paradox: the axioms of $\mathsf{ZFC}$ let us form the set of all ordinals onto which a given set surjects, so by Burali-Forti we know that (according to such a theory) for every set $x$ there is some ordinal $y$ which $x$ does not surject onto.
How do the $\mathsf{ZFC}$-axioms let us do this? Well, given a surjection $x\rightarrow\alpha$ for some ordinal $\alpha$ we can "pull back" to get a (pre)well-ordering of $x$ with ordertype $\alpha$. We can then (pre)well-order the (pre)well-orderings of $x$ according to length, to get a new well-ordering $w$ of a more complicated set (a particular set of sets of binary relations on $X$). This $w$ is "longer" than every ordinal. Now the above is more-or-less uncontroversial, and all the usual set theories (to my knowledge) let you do this; where $\mathsf{ZFC}$ becomes special is when it lets you go further and say that every well-ordering is isomorphic to some ordinal (this is one of the paradigmatic proofs by transfinite recursion), at which point we have a contradiction with Burali-Forti.

Indeed, more can be said (ignoring some issues re: phrasing):

If $x$ is a collection of sets and there is some set $y$ such that there is no surjection from $x$ to $y$, then $x$ is a set.

(This actually seems to use Choice, at least as far as I'm aware; I'll think about whether we can drop Choice in the morning.)
So in $\mathsf{ZFC}$ (and its relatives), size is the only relevant criterion for determining sethood.

An interesting question at this point is whether there are any "size distinctions" to be made among the proper classes. It turns out that this is a very rich topic, and in particular the usual axioms of set theory do not settle the question at all. For more about this, see e.g. the discussion here.
