Axioms of set theory
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. $$\forall a \forall b [\forall x (x \in a \longleftrightarrow x \in b) \rightarrow a =b]$$
