Definition:LAST

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: $$\and$$;
 * The or symbol: $$\or$$;
 * The negation symbol: $$\lnot$$.


 * 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:
 * $$\left({v_n \in v_m}\right)$$
 * $$\left({v_n \in w_m}\right)$$
 * $$\left({w_n \in v_m}\right)$$
 * $$\left({w_n \in w_m}\right)$$
 * $$\left({v_n \in v_m}\right)$$
 * $$\left({v_n \in w_m}\right)$$
 * $$\left({w_n \in v_m}\right)$$
 * $$\left({w_n \in w_m}\right)$$

is a formula of LAST.

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


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