Definition:LAST

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


 * 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.

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


 * If $\phi$ is a formula of LAST, then $\left({\neg \phi}\right)$ is a formula of LAST.

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


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