Material set theory

Axiom of Foundation

The Axiom of Foundation or Axiom of Regularity is a possible axiom of Material set theory first proposed by John von Neumann: zf

which is to say, every set containing at least one set has an element disjoint to itself. The Axiom of Foundation thus precludes¹ infinite chains of elementhood and therefore demands that all sets be well-founded — in fact, this demand turns out to be equivalent to the Axiom of Foundation over the Axiom of Dependent Choice.

Proof

proof

Relation to other axioms

• A strong negation is Aczel’s Antifoundation Axiom

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Footnotes

1. Possibly only with the Axiom of Dependent Choice ↩