Rasiowa–Sikorski lemma


In axiomatic set theory, the Rasiowa–Sikorski lemma is one of the most fundamental facts used in the technique of forcing. In the area of forcing, a subset E of a poset is called dense in P if for any p ∈ P there is e ∈ E with e ≤ p. If D is a family of dense subsets of P, then a filter F in P is called D-generic if
Now we can state the Rasiowa–Sikorski lemma:

Proof of the Rasiowa–Sikorski lemma

The proof runs as follows: since D is countable, one can enumerate the dense subsets of P as D1, D2, …. By assumption, there exists p ∈ P. Then by density, there exists p1 ≤ p with p1 ∈ D1. Repeating, one gets … ≤ p2 ≤ p1 ≤ p with pi ∈ Di. Then G = is a D-generic filter.
The Rasiowa–Sikorski lemma can be viewed as an equivalent to a weaker form of Martin's axiom. More specifically, it is equivalent to MA.

Examples