Lemma from Rasiowa-Sikorski

The Rasiowa-Sikorski lemma , named after the Polish mathematicians Roman Sikorski and Helena Rasiowa , is fundamental to the development of the forcing method in set theory . It ensures the existence of filters with certain properties.

statement

Let be a quasi-order and a at most countable set of dense subsets of . Then there is a filter for each with the properties: ${\displaystyle \langle P,\leq _{P}\rangle }$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle P}$ ${\displaystyle p_{0}\in P}$ ${\displaystyle F\subseteq P}$

• ${\displaystyle p_{0}\in F}$
• ${\displaystyle D\cap F\neq \emptyset }$, for everyone . ${\displaystyle D\in {\mathcal {D}}}$

Filters with the last property are also called generic. ${\displaystyle {\mathcal {D}}}$

proof

Let be an enumeration of the sets in and define for recursive: ${\displaystyle D_{1},D_{2},\dots }$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle n\geq 0}$

${\displaystyle p_{n+1}:=}$"an element with ". ${\displaystyle p\in D_{n+1}}$ ${\displaystyle p\leq p_{n}}$

Such exists due to the tightness of . Then the quantity is such a filter. ${\displaystyle p_{n+1}}$ ${\displaystyle D_{n+1}}$ ${\displaystyle F:=\{p\in P\mid \exists n\in \mathbb {N} :p_{n}\leq p\}}$

Extensions

It can be shown that the statement becomes generally false if the cardinality has. The question whether the lemma holds for cardinal numbers with leads to Martin's axiom . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle 2^{\aleph _{0}}}$ ${\displaystyle \kappa }$ ${\displaystyle \aleph _{0}<\kappa <2^{\aleph _{0}}}$

literature

• Jech, Thomas: Set Theory , Springer-Verlag Berlin Heidelberg (2006), ISBN 3-540-44085-2 .
• Kunen, Keneth: Set Theory: An Introduction to Independence Proofs , North-Holland (1980), ISBN 0-444-85401-0 .