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.
  • 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.


Sources