Rahul Sharma (Editor)

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 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

Contents

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.

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( ℵ 0 ).

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 ℵ 0 and the poset has the countable chain condition, we can instead use Martin's axiom.

    • References

    Rasiowa–Sikorski lemma Wikipedia
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