In axiomatic set theory, the Rasiowa–Sikorski lemma (named after Roman Sikorski

Roman Sikorski

Roman Sikorski was a Polish mathematician.Sikorski was from 1952 until 1982 professor at the Warsaw University...

 and Helena Rasiowa

Helena Rasiowa

Helena Rasiowa was a Polish mathematician. She worked in the foundations of mathematics and algebraic logic.-Early years:...

) is one of the most fundamental facts used in the technique of forcing

Forcing (mathematics)

In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...

. 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

Filter (mathematics)

In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...

 F in P is called D-generic

Generic filter

In the mathematical field of set theory, a generic filter is a kind of object used in the theory of forcing, a technique used for many purposes, but especially to establish the independence of certain propositions from certain formal theories, such as ZFC...

 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

Dense order

In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x...

 subsets of P then there exists a D-generic filter

Filter (mathematics)

In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...

 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 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

Martin's axiom

In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...

. More specifically, it is equivalent to MA().

Examples

• For (P, ≥) = (Func(X, Y), ⊂), the poset of partial function
  Partial function
  In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...
  
  s 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 filter
  Generic filter
  In the mathematical field of set theory, a generic filter is a kind of object used in the theory of forcing, a technique used for many purposes, but especially to establish the independence of certain propositions from certain formal theories, such as ZFC...
  
  s, {H ∪ G₀: P_ijP_t} forms an H-generic filter
  Generic filter
  In the mathematical field of set theory, a generic filter is a kind of object used in the theory of forcing, a technique used for many purposes, but especially to establish the independence of certain propositions from certain formal theories, such as ZFC...
  
  .
• If D is uncountable, but of cardinality strictly smaller than and the poset has the countable chain condition
  Countable chain condition
  In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions: the upwards and downwards countable chain conditions. These are not equivalent...
  
  , we can instead use Martin's axiom
  Martin's axiom
  In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
  
  .

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