Axiomatic Set Theories

ZF (Zermelo-Frankel Set Theory)

Axioms:

Axiom of Extensionality: If the sets X and Y have the same elements, then X =Y
Axiom of Pairing: For any set x, y, then there exists a set {x, y}
Axiom Schema of Separation: Given any set Y and property $φ$ , there is a subset X $\subseteq$ Y which contains exactly the elements in Y which satisfy $φ$
Axiom of Union: For any set X, there exists a set Y = $⋃ X$ , the union of all elements of X (or the set of elements of the sets in X)
Axiom of Powerset: For any set X, there exists a set Y = P(X), which the set of all subsets of X
Axiom of Infinity: There exists an infinite set. Formally, $\exists I (\emptyset \in I \land \forall x (x \in I ⟹ (x \cup x) \in I))$ (a set built this way is called an inductive set)
Axiom Schema of Replacement: If class $F$ is a function, then for any set $X$ there exists a set $Y = F (X) = {F (x) ∣ x \in X}$ , i.e., for any definable-class function the image of a set under this function is also a set
Axiom of Foundation: Every nonempty set has an $\in$ -minimal element (i.e., $x \cap X = \emptyset$ for some $x \in X$ ). Formally, $\forall x (x \neq = \emptyset \to (\exists y \in x) (y \cap x = \emptyset))$

Other axiom(s)

Axiom of Choice: For every indexed family $(S_{i})_{i \in I}$ of nonempty sets, there exists an indexed set $(x_{i})_{i \in I}$ such that $x_{i} \in S_{i}$ for every $i \in I$ . Formally, $\forall X [\emptyset \neq \in X ⟹ \exists f : X \to A \in X ⋃ A \forall A \in X (f (A) \in A)]$ Note: we must choose an element from each set in a family sets $S$ as required by the choice function $f$ , however, if we wish to ‘skip’ a set we merely take a subset $A \subset S$ which does not contain that particular set which we wish to ‘skip.’

$ZF +$ axiom of choice is denoted $ZFC$

NBG (Von Neumann-Bernays-Gödel Set Theory)

axioms:

KP (Kripke-Platek Set Theory)

MK (Morse-Kelly Set Theory)

Properties of Theories