Class (set theory)

The notion of a class is a generalisation of that of a set. Every set is a class, while a class is not necessarily a set. A class which is not a set is called a proper class.

In ZFC set theory
When we work in a sufficiently strong set theory \(T\) such as \(\textrm{ZFC}\) set theory or \textrm{KP} set theory, a class simply means a formula (equipped with a fixed free variable \(\xi\)) in \(T\). When we refer to a class, then we usually denote it by a single capital letter such as \(X\) in order to introduce conventions such as \(a \in X\). Given a class \(X\) in \(T\) and a set \(a\), we abbreviate to \(a \in X\) the result of replacing all free instances of \(\xi\) by \(a\) in \(X\), which is commonly denoted by \(X[a/\xi]\) in first order logic. Since \(T\) itself is unable to directly refer to formulae in \(T\) without Goedel correspondence, quantification of classes is not allowed in this context.

For a set \(x\) and a class \(X\), we abbreviate to \(x = X\) the formula "for any set \(y\), \(y \in x\) is equivalent to \(y \in X\)" in \(T\). In this case, \(X\) is called a set, as if it is a term in \(T\). We abbreviate to \(x \neq X\) the negation of \(x \in X\). The statement "\(X\) is a proper class" means the formula "for any set \(x\), \(x \neq X\)", which we can ask the provability in \(T\).

Given a set \(x\), we also denote by \(x\) the class \(\xi \in x\). The abuse of notation is not critically ambiguous, because \(x = X\) is provable in \(T\), where (X\) denotes the class \(\xi \in x\). In this sense, every set is a class by definition.

Further, for classes \(X\) and \(Y\), we abbreviated to \(X \in Y\) the formula "there exists a set \(x\) such that \(x = X\) and \(x \in Y\)", and to \(X = Y\) the formula "for any set \(x\), \(x \in X\) is equivalent to \(x \in Y\)". In this way, we can extend predicates for sets to predicates for classes.

In NBG set theory
When we work in a set theory \(T\) which is \(\textrm{NBG}\) set theory or its extension sharing the language, a class simply means a term, as a set in \(\textrm{ZFC}\) set theory simply means a term. In particular, quantification of classes is allowed and \(\in\) and \(=\) automatically make sense for classes in this context.

Given a formula \(F\) with a fixed free variable \(\xi\) in \(T\) without unbounded quantification, the statement "there uniquely exists a class \(X\) such that for any set \(x\), \(x \in X\) is equivalent to \(F[x/\xi]\)" is provable in \(T\). In this sense, such a formula in \(T\) can be regarded as a class. If \(T\) is \(\textrm{MK}\) set theory or its extension, then the restriction of the quantification can be dropped.

When \(T\) is two sorted, we have a sort of variables \(x\) called a set. In this case, the statement "\(X\) is a proper class" means "for any set \(x\), \(x \neq X\)". When \(T\) is not sorted, then a set means a class \(x\) for which there exists a class \(X\) such that \(x \in X\). In this case, the statement "\(X\) is a proper class" means the negation of "\(X\) is class", i.e. "for any class \(Y\), \(X \notin Y\)".

Examples
We have many examples of proper classes appearing in googology:
 * The class \(V\) (or \(\textrm{WF}\) when we work in a set theory without the regularity) of sets.
 * The class \(L\) of constructible sets.
 * The class \(\textrm{HOD}\) of hereditarily ordinal definable sets.
 * The class \(\textrm{OD}\) of ordinal definable sets.
 * The class \(\textrm{Card}\) of cardinals.
 * The class \(\textrm{Reg}\) of regular cardinals.
 * The class \(\textrm{On}\) of ordinals.
 * The class \(\textrm{AP}\) of additive principal numbers.