Zermelo Fraenkel Set Theory

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.

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