# Definition:LAST

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## 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.

- Quantifier symbols:
- The
**universal**quantifier:**For all**: $\forall$ - The
**existential**quantifier:**There exists**: $\exists$

- The

- Punctuation symbols:
**Parentheses**: $($ and $)$.

### 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 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**.