# Definition:LAST It has been suggested that this page or section be merged into Definition:Language of Set Theory. (Discuss)

## Definition

LAST stands for LAnguage of Set Theory.

It is a formal system designed for the description of sets.

## Formal Language

This is the formal language of LAST:

### The Alphabet

The alphabet of LAST is as follows:

#### The Letters

The letters of LAST come in two varieties:

• Names of sets: $w_0, w_1, w_2, \ldots, w_n, \ldots$

These are used to refer to specific sets.

• Variables for sets: $v_0, v_1, v_2, \ldots, v_n, \ldots$

These are used to refer to arbitrary sets.

#### The Signs

The signs of LAST are as follows:

• The membership symbol: $\in$, to indicate that one set is an element of another.
• The equality symbol: $=$, to indicate that one set is equal to another.
• Logical connectives:
• The and symbol: $\land$
• The or symbol: $\lor$
• The negation symbol: $\neg$
• Quantifier symbols:

### Formal Grammar

The formal grammar of LAST is as follows:

• Any expression of one of these forms:
• $\left({v_n = v_m}\right)$
• $\left({v_n = w_m}\right)$
• $\left({w_m = v_n}\right)$
• $\left({w_n = w_m}\right)$
• $\left({v_n \in v_m}\right)$
• $\left({v_n \in w_m}\right)$
• $\left({w_m \in v_n}\right)$
• $\left({w_n \in w_m}\right)$

is a formula of LAST.

• If $\phi, \psi$ are formulas of LAST, then:
• $\left({\phi \land \psi}\right)$
• $\left({\phi \lor \psi}\right)$

are formulas of LAST.

• If $\phi$ is a formula of LAST, then $\left({\neg \phi}\right)$ is a formula of LAST.
• If $\phi$ is a formula of LAST, then expressions of the form:
• $\left({\forall v_n \phi}\right)$
• $\left({\exists v_n \phi}\right)$

are formulas of LAST.

• No expressions that can not be constructed from the above rules are formulas of LAST.