Sets and classes

Question

I was just wondering if anyone could tell me the mathematical definitions of the two and how they are different, i seem to be only able to find a partial answer, for example i know that in ZFC that sets are elements of other sets and classes are not elements of any other class, i'm also struggling to think of an example that can illustrate this.

Answer 1

The idea behind the definition of classes is to reflect the intuitions that originally led to the (unrestricted) comprehension axiom, which stated that for any statement in set theory with one free variable, $P(x)$, there was a set $\{x | P(x)\}$ such that $\forall y, y \in \{x | P(x)\} \iff P(y)$. This axiom seemed very natural to the developers of set theory, but it was proved to lead to contradiction via Russell's Paradox. The axiom had to be replaced with that of restricted comprehension (aka separation), i.e. that there are sets $\{x \in Z | P(x)\}$ such that $\forall y, y \in \{x \in Z | P(x)\} \iff (P(y) \wedge y \in Z)$.

The definition of a class is intended as a dumping ground for the intuition that we should be able to form 'sets' of elements that all have some property. Formally, a class is an equivalence class of statements with one free variable, where $P(x)$ is equivalent to $Q(x)$ if we can prove $\forall y, P(y) \iff Q(y)$. The intuition is then that the class associated to $P(x)$ contains all sets $y$ such that $P(y)$ is true. For example, $\underline{Ord}$ is the class associated to the statement (edited) $P(x) = (\forall y,z \in x, (y\in z) \vee (y=z) \vee (z \in y)) \wedge (\forall y,z,w \in x, ((y \in z) \wedge (z \in w)) \implies (y \in w))$.

However, we must always be aware that classes are not sets: i.e. they are not elements of the model of set theory.