Rasiowa–Sikorski lemma

In axiomatic set theory, the Rasiowa–Sikorski lemma (named after Helena Rasiowa and Roman Sikorski) is one of the most fundamental facts used in the technique of forcing. In the area of forcing, a subset E of a forcing notion (P, ≤) 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, a filter F in P is called D-generic if

Now we can state 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 p_i ∈ D_i. Then G = { q ∈ P: ∃ i, q ≥ p_i} is a D-generic filter.

The Rasiowa–Sikorski lemma can be viewed as a weaker form of an equivalent to Martin's axiom. More specifically, it is equivalent to MA(${\displaystyle \aleph _{0}}$).

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