Zermelo–Fraenkel set theory
Formal language

Formally, ZFC is a 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 has a single predicate symbol, usually denoted ∈ {\displaystyle \in }, which is a predicate symbol of arity 2 (a binary relation symbol). This symbol symbolizes a set membership relation. For example, the formula a ∈ b {\displaystyle a\in b} means that a {\displaystyle a} is an element of the set b {\displaystyle b} (also read as a {\displaystyle a} is a member of b {\displaystyle b}).

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:

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

A well-formed formula can be thought as a syntax tree. The leaf nodes are always atomic formulae. Nodes ∧ {\displaystyle \land } and ∨ {\displaystyle \lor } have exactly two child nodes, while nodes ¬ {\displaystyle \lnot }, ∀ x {\displaystyle \forall x} and ∃ x {\displaystyle \exists x} have exactly one. There are countably infinitely many wff, however, each wff has a finite number of nodes.
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