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:
  • 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:
  • $\paren {v_n = v_m}$
  • $\paren {v_n = w_m}$
  • $\paren {w_m = v_n}$
  • $\paren {w_n = w_m}$
  • $\paren {v_n \in v_m}$
  • $\paren {v_n \in w_m}$
  • $\paren {w_m \in v_n}$
  • $\paren {w_n \in w_m}$

is a formula of LAST.

• If $\phi, \psi$ are formulas of LAST, then:
  • $\paren {\phi \land \psi}$
  • $\paren {\phi \lor \psi}$

are formulas of LAST.

• If $\phi$ is a formula of LAST, then $\paren {\neg \phi}$ is a formula of LAST.

• If $\phi$ is a formula of LAST, then expressions of the form:
  • $\paren {\forall v_n \phi}$
  • $\paren {\exists v_n \phi}$

are formulas of LAST.

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

Sources

• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.): $\S 2.1$