# Definition:Language of Set Theory

## Definition

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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:

\(\ds P \implies Q\) | \(\operatorname{for}\) | \(\ds \neg ( P \land \neg Q )\) | ||||||||||||

\(\ds P \lor Q\) | \(\operatorname{for}\) | \(\ds \neg ( \neg P \land \neg Q )\) | ||||||||||||

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

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

\(\ds ( P \land Q )\) | \(\operatorname{for}\) | \(\ds \neg ( P \implies \neg Q )\) | ||||||||||||

\(\ds ( P \lor Q )\) | \(\operatorname{for}\) | \(\ds ( \neg P \implies Q )\) | ||||||||||||

\(\ds ( P \iff Q )\) | \(\operatorname{for}\) | \(\ds \neg ( ( P \implies Q ) \implies \neg ( Q \implies P ) )\) |

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

\(\ds \neg P\) | \(\operatorname{for}\) | \(\ds ( P \mid P )\) | ||||||||||||

\(\ds ( P \implies Q )\) | \(\operatorname{for}\) | \(\ds ( P \mid ( Q \mid Q ) )\) |

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:

- If $x$ and $y$ are variables, then $x \in y$ is a well-formed formula.
- 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.
- 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.

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 2$ - 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Language of Set Theory, Formulas