Set Theory/Zermelo-Fraenkel Axiomatic Set Theory

< Set Theory

The axiomsEdit

  • Extensionality, two sets with the same elements are equal.
\forall x,y,z \ (z \in x \Leftrightarrow z \in y) \Rightarrow (x=y)
  • Separation, subsets exist
\forall y_1,p \ \exists y_2 \  \forall x\  x\in y_2 \Leftrightarrow 
(p \wedge x\in y_1)
where p is any proposition
  • The empty set exists
\exists x \ \forall y\  y\not\in x
  • Union, the union of all members of a set is a set.
\forall x\  \exists  y\  \forall z \ z\in y \Leftrightarrow (\exists u \ z\in u \wedge u\in x)
  • Power sets exist
\forall x \ \exists y \ \forall z \ z\in y \Leftrightarrow  
(\forall t \ t \in z \Rightarrow t \in x)
we denote this set y by P(x)
  • Infinity, an infinite set exists
\exists x \ (\empty \in x) \wedge ( \forall y \ y\in x \Rightarrow P(y)\in x)
  • Foundation, no set is a member of itself
\forall x \ x \ne \empty \Rightarrow (\exists y \in x \ y \cap x = \empty)

Cardinals · Naive Set Theory