Rasiowa-Sikorski lemma

From Wikipedia, the free encyclopedia

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.

Contents 1 Proof of the Rasiowa-Sikorski lemma 2 Examples 3 External links 4 See also 5 References

[edit] Proof of the Rasiowa-Sikorski lemma

The proof runs as follows: because 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(ℵ₀).

[edit] Examples

• For (P, ≥) = (Func(X, Y), ⊂), the poset of partial functions from X to Y, define D_x = {s ∈ P: x ∈ dom(s)}. If X is countable, the Rasiowa-Sikorski lemma yields a {D_x: 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: P_ijP_t} forms an H-generic filter.
• If D is uncountable, but of cardinality strictly smaller than 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

[edit] See also

• forcing
• generic filter
• Martin's axiom

[edit] References

• Ciesielski, K. Set Theory for the Working Mathematician, London Mathematical Society
• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. North-Holland. ISBN 0-444-85401-0.