# Mostowski collapse lemma

In mathematical logic, the **Mostowski collapse lemma**, also known as the **Shepherdson–Mostowski collapse**, is a theorem of set theory introduced by Andrzej Mostowski (1949, theorem 3) and John Shepherdson (1953).

## Statement

Suppose that *R* is a binary relation on a class *X* such that

*R*is set-like:*R*^{−1}[*x*] = {*y*:*y**R**x*} is a set for every*x*,*R*is well-founded: every nonempty subset*S*of*X*contains an*R*-minimal element (i.e. an element*x*∈*S*such that*R*^{−1}[*x*] ∩*S*is empty),*R*is extensional:*R*^{−1}[*x*] ≠*R*^{−1}[*y*] for every distinct elements*x*and*y*of*X*

The Mostowski collapse lemma states that for any such *R* there exists a unique transitive class (possibly proper) whose structure under the membership relation is isomorphic to (*X*, *R*), and the isomorphism is unique. The isomorphism maps each element *x* of *X* to the set of images of elements *y* of *X* such that *y R x* (Jech 2003:69).

## Generalizations

Every well-founded set-like relation can be embedded into a well-founded set-like extensional relation. This implies the following variant of the Mostowski collapse lemma: every well-founded set-like relation is isomorphic to set-membership on a (non-unique, and not necessarily transitive) class.

A mapping *F* such that *F*(*x*) = {*F*(*y*) : *y R x*} for all *x* in *X* can be defined for any well-founded set-like relation *R* on *X* by well-founded recursion. It provides a homomorphism of *R* onto a (non-unique, in general) transitive class. The homomorphism *F* is an isomorphism if and only if *R* is extensional.

The well-foundedness assumption of the Mostowski lemma can be alleviated or dropped in non-well-founded set theories. In Boffa's set theory, every set-like extensional relation is isomorphic to set-membership on a (non-unique) transitive class. In set theory with Aczel's anti-foundation axiom, every set-like relation is bisimilar to set-membership on a unique transitive class, hence every bisimulation-minimal set-like relation is isomorphic to a unique transitive class.

## Application

Every set model of ZF is set-like and extensional. If the model is well-founded, then by the Mostowski collapse lemma it is isomorphic to a transitive model of ZF and such a transitive model is unique.

Saying that the membership relation of some model of ZF is well-founded is stronger than saying that the axiom of regularity is true in the model. There exists a model *M* (assuming the consistency of ZF) whose domain has a subset *A* with no *R*-minimal element, but this set *A* is not a "set in the model" (*A* is not in the domain of the model, even though all of its members are). More precisely, for no such set *A* there exists *x* in *M* such that *A* = *R*^{−1}[*x*]. So *M* satisfies the axiom of regularity (it is "internally" well-founded) but it is not well-founded and the collapse lemma does not apply to it.

## References

- Jech, Thomas (2003),
*Set Theory*, Springer Monographs in Mathematics (third millennium ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7 - Mostowski, Andrzej (1949), "An undecidable arithmetical statement" (PDF),
*Fundamenta Mathematicae*, Institute of Mathematics Polish Academy of Sciences,**36**(1): 143–164, doi:10.4064/fm-36-1-143-164 - Shepherdson, John (1953), "Inner models for set theory, Part III",
*Journal of Symbolic Logic*, Association for Symbolic Logic,**18**: 145–167, doi:10.2307/2268947