Def/Language of set theory



=Grammar= The grammatical axioms of the language of set theory, defining well-formed formulae, are:
 * If "$v$" and "$w$" are variable symbols then "$(v = w)$" is a wff and "$(v \in w)$" is a wff.
 * If "$A$" and "$B$" are wff's, then "$(A \wedge B)$" is a wff.
 * If "$A$" is a wff, then "$(\neg A)$" is a wff.
 * If "$A$" is a wff, and $v$ is a variable symbol, then "$(\forall v A) $" is a wff.

=Basic Enlargements= The list of symbols is frequently enlarged, with commonly used symbols and words. In the following list of abbreviations, $"A"$ and $"B"$ are arbitrary wff's.

$$(\exists v A(v) ) \wedge (\forall v_1 \forall v_2, A(v_1) \wedge A(v_2) \Rightarrow v_1 = v_2)$$
 * English and Greek letters, with numerical subscripts, are used often as variable symbols.
 * $\vee$ ("or") is an abbreviation: "$A \vee B$" stands for "$\neg (\neg A \wedge \neg B)$".
 * $\Rightarrow$ ("implies") is an abbreviation: "$A \Rightarrow B$" stands for "$\neg (A \wedge \neg B)$".
 * $\Leftrightarrow$ ("iff") is an abbreviation: "$A \Leftrightarrow B$" stands for "$(A \Rightarrow B) \wedge (B \Rightarrow A)$".
 * $\not \in$ and $\neq$. "$v \not \in w$" stands for "$\neg (v \in w)$", for example.
 * $\exists$ ("there exists"). "$\exists v A$" stands for "$\neg (\forall v \neg A)$".
 * $\exists!$ ("there exists a unique"). "$\exists ! v A(v)$" stands for:

In addition, other abbreviations are used with quantifiers, especially:
 * "$\exists x \in X, A$" stands for "$\exists x, x \in X \wedge A$".
 * "$\forall x \in X, A$" stands for "$\forall x, x \in X \Rightarrow A$".

=Examples= The sentence "$\forall x x \neq x$" is a wff in the language of set theory. Observe that wff's are not necessarily "true sentences", they are simply "grammatical sentences".

The sentence "$\forall x y \exists =$" is not a wff.

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