Rasiowa–Sikorski lemma

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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 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.

Proof of the Rasiowa–Sikorski lemma[edit]

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[edit]

  • For (P, ≥) = (Func(X, Y), ⊂), the poset of partial functions from X to Y, define Dx = {sP: x ∈ dom(s)}. If X is countable, the Rasiowa–Sikorski lemma yields a {Dx: xX}-generic filter F and thus a function ∪ F: XY.
  • If we adhere to the notation used in dealing with D-generic filters, {HG0: 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.

See also[edit]

References[edit]

External links[edit]

  • 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