# Rasiowa–Sikorski lemma

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 poset (*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*, then a filter

*F*in

*P*is called

*D*-generic if

*F*∩*E*≠ ∅ for all*E*∈*D*.

Now we can state the **Rasiowa–Sikorski lemma**:

## 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 an equivalent to a weaker form of Martin's axiom. More specifically, it is equivalent to MA().

## Examples

- For (
*P*, ≤) = (Func(*X*,*Y*), ⊇), the poset of partial functions from*X*to*Y*, reverse-ordered by inclusion, 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 and the poset has the countable chain condition, we can instead use Martin's axiom.

## See also

## References

- Ciesielski, Krzysztof (1997).
*Set theory for the working mathematician*. London Mathematical Society Student Texts.**39**. Cambridge: Cambridge University Press. ISBN 0-521-59441-3. Zbl 0938.03067. - Kunen, Kenneth (1980).
*Set Theory: An Introduction to Independence Proofs*. Studies in Logic and the Foundations of Mathematics.**102**. North-Holland. ISBN 0-444-85401-0. Zbl 0443.03021.

## 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