eolas/Set Theory/Axioms of set theory.md

tags
Logic Set_Theory theorems-axioms-laws

The basic notions of set theory are defined in Basic properties of sets. There we introduced a formal syntax that will be utilised to define the axioms. For easy reference:

• variables a,b,c,... to range over sets
• variables x,y,z to range over ordinary objects as well as sets.

Axiom of Extensionality

Sets which contain the same members are the same set. If sets A and B contain the same elements then A = B.

\forall a \forall b [\forall x (x \in a \longleftrightarrow x \in b) \rightarrow a =b]