Overview
Zermelo Fraenkel is one of the primary versions of set theory, and is named after the mathematicians Ernst Zermelo and Abraham Fraenkel who initially developed the theory.
Axioms
- Existence
{% \exists x (x=x) %}
- Extensionality
{% \forall x \forall y [\forall z (z \in x \iff z \in y) \rightarrow x=y] %}
- Foundation
{% \forall x [\exists y (y \in x) \rightarrow \exists y (y \in x \;and \; \neg \exists z (z \in x \land z \in y))] %}
- Comprehension
{% \exists y \forall x (x \in y \iff x \in z \land \phi) %}
- Pairing
{% \forall x \forall y \exists z ((x \in z)\land(y \in z)) %}
- Union
{% \forall F \exists A \forall Y \forall x (x \in Y \land Y \in F \rightarrow x \in A) %}
- Replacement
{% \forall x \in A \exists! y \phi(x,y) \rightarrow \exists Y \forall x \in A \exists y \in Y \phi(x,y) %}
- Infinity
{% \exists x (0 \in x \land \forall y \in x (S(y) \in x)) %}
- Power Set
{% \forall x \exists y \forall z (z \subset x \rightarrow z \in y) %}
Axiom of Choice
The Zermelo Fraenkel set theory is usually extended to include the Axiom of Choice, in which it is then referred to as ZFC.