Set Theory

Axioms

Axiom 1 (Axiom of Extension). Two sets are equal if and only if they have the same elements.

Axiom 2 (Axiom of Specification). To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly those elements $x$ of $A$ for which $S(x)$ holds.

Axiom 3 (Axiom of Pairing). For any two sets there exists a set that they both belong to.

Axiom 4 (Axiom of Unions). For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection.

Axiom 5 (Axiom of Powers). For each set there exists a collection of sets that contains among its elements all the subsets of the given set.

Axiom 6 (Axiom of Infinity). There exists a set containing 0 and containing the successor of each of its elements.

Axiom 7 (Axiom of Choice). The Cartesian product of a non-empty family of non-empty sets is non-empty.

Axiom 8 (Axiom of Substitution). If $S(a, b)$ is a sentence such that for each $a$ in a set $A$ the set ${b : S(a, b)}$ can be formed, then there exists a function $F$ with domain $A$ such that $F(a) = {b : S(a, b)}$ for each $a$ in $A$.