Definition:LAST

It has been suggested that this page or section be merged into Definition:Language of Set Theory. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mergeto}} from the code.

This page has been identified as a candidate for refactoring of advanced complexity. Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

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.

• 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

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