Editing Zermelo–Fraenkel set theory (section)

== Formal language ==
{{See also|Formal language}}

Formally, ZFC is a [[many-sorted first-order logic|one-sorted theory]] in [[first-order logic]]. The equality symbol can be treated as either a primitive logical symbol or a high-level abbreviation for having exactly the same elements. The former approach is the most common. The [[signature (mathematical logic)|signature]] has a single predicate symbol, usually denoted <math>\in</math>, which is a predicate symbol of arity 2 (a binary relation symbol). This symbol symbolizes a [[set membership]] relation. For example, the [[Well-formed formula|formula]] <math>a\in b</math> means that <em><math>a</math> is an element of the set <math>b</math></em> (also read as <em><math>a</math> is a member of <math>b</math></em>).

There are different ways to formulate the formal language. Some authors may choose a different set of connectives or quantifiers. For example, the logical connective NAND alone can encode the other connectives, a property known as [[functional completeness]]. This section attempts to strike a balance between simplicity and intuitiveness.

The language's alphabet consists of:

* A countably infinite amount of variables used for representing sets
* The logical connectives <math>\lnot</math>, <math>\land</math>, <math>\lor</math>
* The quantifier symbols <math>\forall</math>, <math>\exists</math>
* The equality symbol <math>=</math>
* The set membership symbol <math>\in</math>
* Brackets ( )

With this alphabet, the recursive rules for forming [[well-formed formulae]] (wff) are as follows:

* Let <math> x </math> and <math> y </math> be [[metavariable]]s for any variables. These are the two ways to build [[atomic formula|atomic formulae]] (the simplest wffs):
:<math>x=y</math>
:<math>x \in y</math>

* Let <math> \phi </math> and <math> \psi </math> be metavariables for any wff, and <math> x </math> be a metavariable for any variable. These are valid wff constructions:
:<math>\lnot \phi </math>
:<math>( \phi \land \psi )</math>
:<math>( \phi \lor \psi )</math>
:<math> \forall x  \phi </math>
:<math> \exists x  \phi </math>

A well-formed formula can be thought as a syntax tree. The leaf nodes are always atomic formulae. Nodes <math> \land </math> and <math> \lor </math> have exactly two child nodes, while nodes <math>\lnot </math>, <math> \forall x  </math> and <math> \exists x  </math> have exactly one. There are countably infinitely many wffs, however, each wff has a finite number of nodes.