The following axioms are the basis of _axiomatic set theory_ - more specifically the axioms here are the basis of axiomatic set theory referred to as _Zermelo-Fraenkel_ (ZFC) set theory. The 'C' in ZFC stands for "choice," which is in reference to the [axiom of choice,](Axioms%20of%20set%20theory.md#Axiom%20of%20Choice) which is omitted by some authors. When this case we simply refer to ZF set theory. These axioms are developed in oder to provide a way of filling the [holes](Paradoxes%20in%20naive%20set%20theory.md) in [naive set theory](Sets%20(index).md#Naive%20and%20axiomatic%20set%20theories) and they are not the final word on axiomitizing set theory since there are also [extensions](Axioms%20of%20set%20theory.md#Extensions%20of%20ZFC%20set%20theory) that are widely used. A key feature of ZFC is that it only formalizes the existance of one mathematical object: [sets.](Sets.md) %%This is given in page 5 of Jech.%% # Axiom of extension For sets $A$ and $B$ _axiom of extension_ states that two sets are equal if and only if every element of set $A$ is an element of set $B.$ We denote this equality as $A=B.$ If the sets are not equal we then write $A\neq B.$ ^aaa68a ([... see more](Axiom%20of%20extension)) # Axiom of separation Consider a set $A$ with elements $x.$ Every _condition_ $S(x)$ corresponds with a set $B$. We denot this as $B=\{x\in A: S(x)\}.$^19c6bb ([... see more](Axiom%20of%20separation)) # Axiom of union ([... see more](Axiom%20of%20union)) # Axiom of pairing ([... see more](Axiom%20of%20pairing)) # Sum axiom ([... see more](Axiom%20of%20sums)) # Axiom of Powers ([... see more](Axiom%20of%20Powers.md)) # Axiom of Choice ([... see more](Axiom%20of%20choice)) # Extensions of ZFC set theory ## Conservative extensions of ZFC [von Neumann-Bernays-Gödel set theory](von%20Neumann-Bernays-Gödel%20set%20theory) ([... see more](Conservative%20extensions%20of%20ZFC)) #MathematicalFoundations/SetTheory