Rasiowa-Sikorski lemma

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".

Proof of the Rasiowa-Sikorski lemma

The proof runs as follows: since "D" is countable, one can enumerate the dense subsets of "P" as "D"₁, "D"₂, …. By assumption, there exists "p" ∈ "P". Then by density, there exists "p"₁ ≤ "p" with "p"₁ ∈ "D"₁. Repeating, one gets … ≤ "p"₂ ≤ "p"₁ ≤ "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($aleph\_0$).

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" ∪ "G"₀: "P"_"ij""P"_"t"} 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.

External links

* Tim Chow's newsgroup article [http://www-math.mit.edu/~tchow/mathstuff/forcingdum Forcing for dummies] is a good introduction to the concepts and ideas behind forcing; it covers the main ideas, omitting technical details

See also

* forcing
*generic filter
*Martin's axiom

References

* Ciesielski, K. "Set Theory for the Working Mathematician", London Mathematical Society
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