Rasiowa–Sikorski lemma

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

Statement

Let be a quasi-ordering, and a countable set of dense subsets of. Then for each of a filter with the features:

• , For all.

Filter with the last property are called - generic.

Evidence

Be an enumeration of quantities in and define recursively for:

" an element ".

Such a exists because of the tightness of. Then the set such a filter.

Extensions

It can be shown that the statement is generally false if the cardinality. The question of whether the lemma is true for cardinal numbers, leading to Martin's axiom.