ZFC Set Theory

• The Zermelo-Fraenkel Set Theory with Axiom of Choice is the most commonly accepted axiomatic schema for Set Theory.

  Here, all elements of a set are also treated as sets.

A1: Axiom of Extensionability §

• If every element of the set is an element of , and every element of the set is an element of the set , then they are equal.

• This gives us a formal reason as to why we can compare sets (i.e., we assume that it is possible to do this).

A2: Axiom of Regularity §

• All non-empty sets contain an element that is disjoint with them.

A3: Axiom Schema of Specification §

• A subset for any restriction defined by some predicate formula always exists.

• Let be a pure formula of Axiomatic Set Theory and let be sets. Then for any set , there exists a set which consists of those elements for which holds.

• Essentially, there is a subset of whose members are precisely the members of that satisfy the formula .

A4: Axiom of Pairing §

• If are sets then there exists a set which contains and as elements.

A5: Axiom of Union §

• The union over the elements of a set exists.

A6: Axiom of Infinity §

• There exists at least one infinite set namely the set containing all Natural Numbers

A7: Axiom of Power Set §

• The power set of a set exists for all sets

A8: Axiom of Choice §

• For any set of nonempty sets , there exists a choice function that is defined on and which maps each set of to an element of the set.

• In other words, we can choose one element from each set within a collection of sets. It asserts that such a choice is nonempty, even if the collection is infinite.