Mostowski collapse lemma

The Mostowski collapse (also: Mostowski'scher isomorphism ) is a set of set theory, which was first formulated in 1949 by the Polish mathematician Andrzej Mostowski. It is an important tool particularly for the construction of models.

Definition

Be a two-digit well-founded relation on a class. About well-founded recursion for defining the transitive collapse by: .

Then for the picture:

• Is a transitive class.

If, in addition extensional, that is, if for all, so also shall continue to apply:

• Is bijective
• .

Therefore represents an isomorphism between the structures and represents and is the only transitive set which ( with the relation ) must be isomorphic.

Examples

• Be the set of odd numbers, and the usual order. Then is well-founded and extensional. It is true: and. Every odd number is thus mapped to the smallest " still free " natural number. Hence the name " collapse".
• Is a well-ordering on, then the order type, so is the unique ordinal which is to is order. The Mostowski collapse can thus be viewed as a generalization of Ordinalzahldefinition.
• Let be a partial order, and a filter. Define the ( well-founded ) by relation: . Is a countable transitive model of ZFC and is also - generic, so defined, the collapse of the model, which plays a fundamental role in the forcing method.