Mostowski’s collapsing lemma states that any extensional well-founded relation is isomorphic to a (necessarily unique) transitive set.
Mostowski’s lemma can be proven in ZF set theory using the axiom of replacement and axiom of separation. In set theories that are not powerful enough to prove the lemma, it can be adopted as a separate axiom; in this case it is sometimes called Mostowski’s principle or axiom beta.
Mostowski’s principle is sort of a “dual” of the axiom of foundation that the membership relation is well-founded (and, by the axiom of extensionality, extensional). A related principle for structural set theory is the axiom of well-founded materialization.
Michael Shulman (2018). Comparing material and structural set theories [arXiv:1808.05204]
Emanuele Frittaion, Giorgio G. Genovesi?: Axiom Beta Implies Elementary Transfinite Recursion [arXiv:2603.13913]
Last revised on September 22, 2026 at 03:58:15. See the history of this page for a list of all contributions to it.