Is it legitimate to quantify over all sets?

Question

This is a bit of a philosophical question, but is it legitimate to quantify over all sets, since the totality of sets is not itself a set? We can certainly quantify over all natural numbers, or all real numbers, or all subsets of real numbers, because there certainly is a set of all naturals, all reals, and all subsets of reals, respectively. So, my question is, can one actually quantify over all sets, or all ordinals, or all cardinals, etc?

Answer 1

Absolutely. The limitation / restriction is on set construction, not on logical propositions.

It is valid and useful to state and prove statements like

For every set $A$, there exists an injective function from $A$ to $\mathcal{P}(A)$.

Or very formally the same thing,

$$ \forall A. \exists f. \forall x \in A. \exists y \in \mathcal{P}(A). ((x,y) \in f \land \forall z.(((x,z)\in f \rightarrow z=y) \land ((z,y) \in f \rightarrow z=x))) $$

What we can't have in ZF(C) set theory is just sets of arbitrary sets, except as allowed by the power set axiom and pair axiom. But even though, for example, there is no set of all sets bijective with the natural numbers, we can still say and prove things about "any set which has a bijection with the natural numbers".

Answer 2

Yes. We usually formalize set theory as first-order logic (with a relation symbol $\in$ and certain axioms involving $\in$ that establish its meaning), so quantifiers always range over the whole domain of discourse (the universe of all sets). Statements with restricted quantifiers like "for all $x\in\Bbb R$, ..." are actually just quantified conditional statements like "for all $x$, if $x\in\Bbb R$, then ...".

You might be confusing quantification (the syntax for forming a sentence by ranging over a domain) with comprehension (an axiom schema asserting the existence of a set formed by ranging over a domain).

Answer 3

Since you characterized your question as philosophical, I'll answer it from the point of view of the relevant philosophical literature. There is an active literature on this question, and the question isn't really settled. One thing basically everyone agrees on is that arguments like the one you give do not motivate rejecting the legitimacy of absolutely general quantification over sets. I take it your argument proceeds as follows:

1. Given any things, if a quantifier ranges over exactly those things, then there must be a set containing exactly those things.
2. There is no set containing exactly all the sets.
3. Therefore, no quantifier ranges over exactly all the sets.

This argument from (1) and (2) to (3) is valid on standard assumptions, and (2) is provable in ZF set theory (and other set theories; exceptions include W.V.O. Quine's New Foundations), but it's very hard to see why (1) would be true. In particular, it's hard to see what could motivate accepting (1) when the cost is giving up an absolutely general theory of sets.

There are related arguments people make in the literature whose assessment is still up for grabs. Generally speaking, making these arguments involves arguing that we should accept as permissive a theory about set-formation as possible. The move people make here is to grant that there is no set that contains every set, in the current sense of "every" (given a context of utterance, which we'll call "C"), but that it is possible to adopt a more expansive meaning of the quantifiers on which it would be true to say that there is a set which contains exactly those things quantified over in C (of course, this set would not be a universal set relative to the new meaning of the quantifiers). A recent defense of this view is J.P. Studd's book Everything, More or Less, and the first chapter of that book is a good overview of the state of the literature as of 2019 (the year it was published).