# set theory

<mathematics> A mathematical formalisation of the theory of "sets" (aggregates or collections) of objects ("elements" or "members"). Many mathematicians use set theory as the basis for all other mathematics. Axiomatic set theory The study of formal systems whose theorems, on the intended interpretation, are the truths of set theory.

Cantorian set theory Set theory in which either the generalized continuum hypothesis or the axiom of choice is an axiom.

Constructible set theory Set theory limited to sets whose existence is assured by the axioms of restricted set theory (see below). In 1938 Goedel proved that the axiom of choice, continuum hypothesis, and generalized continuum hypothesis are theorems (even if not axioms) of constructible set theory.

Non-Cantorian set theory Set theory in which either the negation of the generalized continuum hypothesis (GCH) or the negation of the axiom of choice (AC) is an axiom. Since GCH => AC, if ~AC is an axiom, then ~GCH will be a theorem.

Restricted set theory standard set theory minus the axiom of choice. Goedel proved in 1938 that if restricted set theory is consistent, then it remains consistent when the axiom of choice is added (and also when the continuum hypothesis is added).

Standard set theory The formal system first formulated by Ernst Zermelo and Abraham Frankel. Also called Zermelo-Frankel set theory or ZF.

Mathematicians began to realise towards the end of the 19th century that just doing "the obvious thing" with sets led to embarrassing paradoxes, the most famous being Russell's Paradox. As a result, they acknowledged the need for a suitable axiomatisation for talking about sets. Numerous such axiomatisations exist; the most popular among ordinary mathematicians is Zermelo Fr"nkel set theory.

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