In ZFC, a class is typically defined as the collection of sets $x$ that satisfy some formula $\varphi(x, p_1, \ldots, p_n)$ for parameters $p_1, \ldots, p_n$. We therefore have that all sets are classes and there is a class of all sets etc. But since there are only countably many formulas, it seems that "most" "collections" of sets are still not classes (for some naive notion of "most" and "collection"). Is this correct?
Many results in ZFC are stated (only) for classes. Take for example the transfinite induction theorem: If $C$ is a class of ordinals such that $\alpha \in C$ implies $\alpha + 1 \in C$, and for limit ordinals $\alpha$ with $\forall \beta < \alpha \ \beta \in C$, we have $\alpha \in C$, then $C$ is the class of all ordinals.
This raises two questions: Is there a precise way to define a notion of a collection of sets such that "everything" is a collection? If so, is it consistent with ZFC that there is such a collection of ordinals that satisfies the two conditions of transfinite induction, but is not the collection of all ordinals?