# Definition:Language of Set Theory

## Definition

The language of set theory consists of the language of predicate logic with the binary predicate symbol $\in$, denoting membership.

### Predicate Symbols

The language of set theory uses only one predicate symbol, $\in$, the membership sign.

It is a binary predicate symbol.

Using $\in$, other symbols such as $=$ can be defined (see Definition:Set Equality).

### Connectives

The language of set theory borrows the connectives from the language of predicate logic. However, some of the connectives can be considered definitional abbreviations.

$\land$ and $\neg$ can be taken as primitive connectives with the following definitional abbreviations:

 $\displaystyle P \implies Q$ $\operatorname{for}$ $\displaystyle \neg ( P \land \neg Q )$ $\displaystyle P \lor Q$ $\operatorname{for}$ $\displaystyle \neg ( \neg P \land \neg Q )$ $\displaystyle P \iff Q$ $\operatorname{for}$ $\displaystyle ( ( P \implies Q ) \land ( Q \implies P ) )$ justified because $\implies$ is already a definitional abbreviation

$\implies$ and $\neg$ can be taken as primitive connectives with the following definitional abbreviations:

 $\displaystyle ( P \land Q )$ $\operatorname{for}$ $\displaystyle \neg ( P \implies \neg Q )$ $\displaystyle ( P \lor Q )$ $\operatorname{for}$ $\displaystyle ( \neg P \implies Q )$ $\displaystyle ( P \iff Q )$ $\operatorname{for}$ $\displaystyle \neg ( ( P \implies Q ) \implies \neg ( Q \implies P ) )$

The Sheffer stroke $\mid$ can be taken as a sole primitive connective with the following definitional abbreviations:

 $\displaystyle \neg P$ $\operatorname{for}$ $\displaystyle ( P \mid P )$ $\displaystyle ( P \implies Q )$ $\operatorname{for}$ $\displaystyle ( P \mid ( Q \mid Q ) )$

The other connectives can be defined using $\neg P$ and $( P \implies Q )$ as the "new" primitive connectives.

### Quantifiers

The language of set theory adopts the same quantifiers as those in the language of predicate logic.

However, only $\forall$ is necessary to adopt as a primitive symbol, and $\exists$ can be defined:

$\exists x: P(x) \operatorname{for} \neg \forall x: \neg P(x)$

## Rules of Formation

The language of set theory is endowed with the following rules of formation:

1. If $x$ and $y$ are variables, then $x \in y$ is a well-formed formula.
2. If $P$ and $Q$ are well-formed formulae, then $\neg P$, $( P \implies Q )$, $( P \land Q )$, $( P \lor Q )$, and $( P \iff Q )$ are also well-formed formulae.
3. If $P$ is a well-formed formula and $x$ is a variable, then $\forall x: ( P )$ and $\exists x: ( P )$ are well-formed formulae.

It is seen that these rules constitute a bottom-up grammar.