Rasiowa–Sikorski lemma

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

FE ≠ ∅ for all ED.

Now we can state the Rasiowa–Sikorski lemma:

Let (P, ≤) be a poset and pP. If D is a countable family of dense subsets of P then there exists a D-generic filter F in P such that pF.

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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 pP. Then by density, there exists p1p with p1D1. Repeating, one gets … ≤ p2p1p with piDi. Then G = { qP: ∃ i, qpi} 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(\aleph_0).

Examples

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