Editing Class (set theory) (section)

== Classes in formal set theories ==
[[ZF set theory]] does not formalize the notion of classes, so each formula with classes must be reduced syntactically to a formula without classes.<ref>{{citation |url=http://us.metamath.org/mpegif/abeq2.html |title=abeq2 – Metamath Proof Explorer |publisher=us.metamath.org |date=1993-08-05 |access-date=2016-03-09}}</ref>  For example, one can reduce the formula <math>A = \{x\mid x=x \}</math> to <math>\forall x(x\in A\leftrightarrow x=x)</math>. For a class <math>A</math> and a set variable symbol <math>x</math>, it is necessary to be able to expand each of the formulas <math>x\in A</math>, <math>x=A</math>, <math>A\in x</math>, and <math>A=x</math> into a formula without an occurrence of a class.<ref>J. R. Shoenfield, "Axioms of Set Theory". In ''Handbook of Mathematical Logic'', Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)</ref><sup>p. 339</sup>

Semantically, in a [[metalanguage]], the classes can be described as [[equivalence classes]] of [[Well-formed formula|logical formulas]]: If <math>\mathcal A</math> is a [[structure (mathematical logic)|structure]] interpreting ZF, then the object language "class-builder expression" <math>\{x \mid \phi \}</math> is interpreted in <math>\mathcal A</math> by the collection of all the elements from the domain of <math>\mathcal A</math> on which <math>\lambda x\phi</math> holds; thus, the class can be described as the set of all predicates equivalent to <math>\phi</math> (which includes <math>\phi</math> itself). In particular, one can identify the "class of all sets" with the set of all predicates equivalent to <math>x=x</math>.{{citation needed|date=June 2024}}<!--Is this treatment of classes common, and does it account for formulas with parameters?-->

Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an [[inaccessible cardinal]] <math>\kappa</math> is assumed, then the sets of smaller rank form a model of ZF (a [[Grothendieck universe]]), and its subsets can be thought of as "classes".

In ZF, the concept of a [[function (mathematics)|function]] can also be generalised to classes. A class function is not a function in the usual sense, since it is not a set; it is rather a formula <math>\Phi(x,y)</math> with the property that for any set <math>x</math> there is no more than one set <math>y</math> such that the pair <math>(x,y)</math> satisfies <math>\Phi</math>. For example, the class function mapping each set to its powerset may be expressed as the formula <math>y = \mathcal P(x)</math>. The fact that the ordered pair <math>(x,y)</math> satisfies <math>\Phi</math> may be expressed with the shorthand notation <math>\Phi(x)=y</math>.

Another approach is taken by the [[von Neumann–Bernays–Gödel axioms]] (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the class existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be a [[conservative extension]] of ZFC.

[[Morse–Kelley set theory]] admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its class existence axioms. This causes MK to be strictly stronger than both NBG and ZFC.

In other set theories, such as [[New Foundations]] or the theory of [[semiset]]s, the concept of "proper class" still makes sense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theory with a [[universal set]] has proper classes which are subclasses of sets.