Axiom:Axiom of Foundation (Classes)

From ProofWiki
Jump to navigation Jump to search


For any non-empty class, there is an element of the class that shares no element with the class.

$\forall S: \neg \paren {S = z: \forall y: \paren {\neg \paren {y \in z} } } \implies \exists x \in S: \neg \paren {\exists w: w \in S \land w \in x}$

Also known as

Otherwise known as the Axiom of Regularity.

Also defined as

It can also be stated as:

A class contains no infinitely descending (membership) sequence. (only equivalent with AoC)
A class contains a (membership) minimal element.