Rasiowa-Sikorski lemma

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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 D of a forcing notion (P, ≤) is called dense in P if for any p ∈ P there is d ∈ D with d ≤ p. A filter F in P is called D-generic if

F ∩ E ≠ ∅ for all E ∈ D.

Now we can state the Rasiowa-Sikorski lemma:

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

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[edit] 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 = { q ∈ P: ∃ i, q ≥ pi} 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).

[edit] Examples

  • For (P, ≥) = (Func(X, Y), ⊂), the poset of partial functions from X to Y, define Dx = {s ∈ P: x ∈ dom(s)}. If X is countable, the Rasiowa-Sikorski lemma yields a {Dx: x ∈ X}-generic filter F and thus a function ∪ F: X → Y.
  • If we adhere to the notation used in dealing with D-generic filters, {H ∪ G0: PijPt} forms an H-generic filter.
  • If D is uncountable, but of cardinality strictly smaller than 2^{\aleph_0} and the poset has the countable chain condition, we can instead use Martin's axiom.

[edit] External links

  • Tim Chow's newsgroup article Forcing for dummies is a good introduction to the concepts and ideas behind forcing; it covers the main ideas, omitting technical details

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[edit] References