Zermelo-Fraenkel Set Theory

Axiom: Axiom of Foundation (Ernst Zermelo, 1908)

Each (regressive) chain of sets \(X_1,X_2,X_3\ldots\), in which each set is a subset of a proceeding set \(X_1\supset X_2 \supset X_3 \supset \ldots \), will end at a finite index \(n\) at a root subset \(X_n\). In other words, every non-empty set \(X\) contains an element that is disjoint from \(X\).

\[\forall X(X\neq\emptyset \Rightarrow\exists (Y\in X)X\cap Y=\emptyset).\]

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