Mostowski collapse

The Mostowski collapse (also: Mostowski's isomorphism theorem) is a theorem from set theory that was first formulated in 1949 by the Polish mathematician Andrzej Mostowski . It is an important aid , especially when constructing models .

definition

So-called collapse of the odd to the natural numbers

Let be a two - digit well-founded relation on a class . About well-founded recursion defines for the transitive collapse by: . ${\displaystyle \prec }$ ${\displaystyle C}$ ${\displaystyle x\in C}$ ${\displaystyle \pi (x)=\{\pi (z)\mid z\prec x\}}$

The following then applies to the figure : ${\displaystyle \pi :C\rightarrow \pi (C)}$

• ${\displaystyle x\prec y\Rightarrow \pi (x)\in \pi (y)}$
• ${\displaystyle \pi (C)}$is a transitive class .

If, in addition, it is extensional, that is, if from already follows for all , then the following also applies: ${\displaystyle E}$ ${\displaystyle \{z\mid z\prec x\}=\{z\mid z\prec y\}}$ ${\displaystyle x=y}$ ${\displaystyle x,y\in C}$

• ${\displaystyle \pi }$is bijective
• ${\displaystyle x\prec y\Leftrightarrow \pi (x)\in \pi (y)}$.

${\displaystyle \pi }$thus represents an isomorphism between the structures and , and is the only transitive set which (with the relation ) is too isomorphic. ${\displaystyle \langle C,\prec \rangle }$ ${\displaystyle \langle \pi (C),\in \rangle }$ ${\displaystyle \pi (C)}$ ${\displaystyle \in }$ ${\displaystyle \langle C,\prec \rangle }$

Examples

• Let be the set of odd numbers and the usual order. Then it is well-founded and extensional. The following applies: and . Every odd number is mapped to the smallest free natural number. Hence the name collapse. ${\displaystyle C=\{1,3,5,\dots \}}$ ${\displaystyle {\prec }={<}}$ ${\displaystyle \prec }$ ${\displaystyle \pi (C)=\omega }$ ${\displaystyle \pi (2n+1)=n}$
• If a well-order is on , then the order type is from , i.e. the uniquely determined ordinal number that is order isomorphic . The Mostowski collapse can thus be viewed as a generalization of the ordinal number definition. ${\displaystyle <}$ ${\displaystyle C}$ ${\displaystyle \pi (C)}$ ${\displaystyle (C,<)}$ ${\displaystyle (C,<)}$
• Be a partial order and a filter . Defining the (well-founded) Relation by: . If ZFC is a countable transitive model and is also generic, then the collapse of defines the model that plays a fundamental role in the forcing method. ${\displaystyle P}$ ${\displaystyle G\subset P}$ ${\displaystyle \in _{G}}$ ${\displaystyle x\in _{G}y\Leftrightarrow \exists p\in G\colon (x,p)\in y}$ ${\displaystyle M}$ ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle \langle M,\in _{G}\rangle }$ ${\displaystyle \langle M[G],\in \rangle }$

literature

• Mostowski, Andrzey: An undecidable arithmetical statement , Fundamenta Mathematicae 36 (1949).
• Jech, Thomas: Set Theory , Springer-Verlag Berlin Heidelberg (2006), ISBN 3-540-44085-2 .