Does Second-Order Comprehension make second-order ZFC inconsistent due to Russell's Paradox?

Question

When we do set theory, we take our first-order variables to range over all sets. But if we take our second-order variables to range over sets of sets in the range of the first-order variables, then second-order comprehension will fail. A simple instance of second-order comprehension such as $\exists X\forall y (Xy \longleftrightarrow y \not\in y)$ will be false on account of Russell’s paradox, according to which no set contains all and only those sets that are not members of themselves.

This doesn't seem correct, but I don't see how it can be avoided without bringing in a class/set distinction which isn't in the axiom as presented in shaprio 1991 for example.

Answer 1

In Second-order Logic, the comprehension schema (considering for simplicity only unary predicate variables) is:

$∃X∀x [ \varphi(x) ↔ X(x) ]$,

where $x$ is an individual variable, $X$ is a unary predicate variable and $X$ does not occur free in $\varphi$.

What prevents form generating inconsistencies ?

The proviso that $X$ is not free in $\varphi$ prevents from using $\lnot X(x)$ to get:

$∃X∀x [ \lnot X(x) ↔ X(x) ]$.

Answer 2

In second-order set theory the second-order quantifier is quantifying classes. And indeed in $\sf ZFC$ there exists a class of exactly those sets which are not members of themselves: it's called the universe.

Also, it might be worth pointing out that second-order $\sf ZF$ is considered with a second-order Replacement axiom schema, rather than a Comprehension schema.

The idea is to compare this to "other" worlds where we use second-order logic. Over the natural numbers the second-order is about sets of numbers, not numbers, namely a collection of numbers; over the real numbers we quantify over a collection of real numbers; in topology we quantify over a collection of open sets, and so on.

So in set theory, the second-order quantification is over collections of objects, which are called classes since the first-order objects are called sets.