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 {{harvs|txt|authorlink=Andrzej Mostowski|first=Andrzej|last= Mostowski|year=1949|loc=theorem 3}} and {{harvs|txt|authorlink=John C. Shepherdson|first=John|last= Shepherdson|year=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 every 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

{{Citation | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | publisher=Springer-Verlag | location=Berlin, New York | edition=third millennium | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003}}
{{citation|title=An undecidable arithmetical statement

|first=Andrzej|last= Mostowski | authorlink = Andrzej Mostowski

|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm36/fm36120.pdf

|publisher= Institute of Mathematics Polish Academy of Sciences

|journal= Fundamenta Mathematicae

|year= 1949

|volume =36

|issue =1

|pages= 143–164

|doi=10.4064/fm-36-1-143-164

|doi-access=free

}}