Axioms of ZFC

* Definition of set by Georg Cantor
Unter einer Menge verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten in unserer Anschauung order unseres Denkens (welche die Elemente von M genannt werden) zu einem ganzen.
(A set is a collection into a whole of definite, distinct objects of our intuition or our thought. The objects are called elements (members) of the set.)

* ZFC : the Zermelo-Fraekel set theory with Choice
The axiom of existence: There exists a set which has no elements.
(empty set ∅ exists)

The axiom of extensionality: If every element of X is an element of Y and every element of Y is an element of X, then X = Y.
(2 sets are identical if they have same elements)

The axiom schema of comprehension: Let P(x) be a property of x. For any A, there is B such that x ∈ A and P(x) holds.
( {x | x ∈ A and P(x)} exists )

The axiom of pair: For any A and B, there is C such that x ∈ C if and only if x = A or x = B.
(C = {A, B} = unordered pair of A and B)

The axiom of union: For any S, there is U such that x ∈ U if and only if x ∈ A for some A ∈ S.
(ex: for S = {A, B}, there is ∪S such that x ∈ ∪{A, B} iff x ∈ A or x ∈ B. ∪S = ∪{A, B} = A ∪ B)

The axiom of power set: For any S, there is P such that X ∈ P if and only if X ⊆ S.

The axiom of infinity: An inductive set exists.
( I is inductive when (a) 0 ∈ I and (b) if n ∈ I, then (n + 1) ∈ I.
0 is the empty set ∅, by definition.
n + 1 is the successor of n defined as n ∪ {n})

The axiom schema of replacement: Let P(x, y) be a property such that for every x there is a unique y for which P(x, y) holds. For every A there is B such that for every x ∈ A there is y ∈ B for which P(x, y) holds.

The axiom of choice: Every system of sets has a choice function.
(choice function (or selector) of a set S is a function defined on S such that g(X) ∈ X for every nonempty X ∈ S.
For S = {{1, 2}, {2, 3, 4}, ∅}, g = { ({1,2}, 1), ({2, 3, 4}, 2), (∅, ∅)} is a selector of S).