axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

, the Rasiowa–Sikorski lemma named after

Helena Rasiowa Helena Rasiowa (20 June 1917 – 9 August 1994) was a Polish mathematician. She worked in the foundations of mathematics and algebraic logic. Early years Rasiowa was born in Vienna on 20 June 1917 to Polish parents. As soon as Poland regained i ...

and

Roman Sikorski Roman Sikorski (July 11, 1920 – September 12, 1983) was a Polish mathematician. Biography Sikorski was a professor at the University of Warsaw from 1952 until 1982. Since 1962, he was a member of the Polish Academy of Sciences. Sikorski's ...

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 set 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: :Let (''P'', ≤) be a

poset In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...

and ''p'' ∈ ''P''. If ''D'' is a

countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...

set of

dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...

subsets of ''P'' then there exists a ''D''-generic filter ''F'' in ''P'' such that ''p'' ∈ ''F''.

Proof of the Rasiowa–Sikorski lemma

Let ''p'' ∈ ''P'' be given. Since ''D'' is countable, ''D'' = , i.e., one can enumerate the dense subsets of ''P'' as ''D''₁, ''D''₂, ... and, by density, there exists ''p''₁ ≤ ''p'' with ''p''₁ ∈ ''D''₁. Iterating that, one gets ... ≤ ''p''₂ ≤ ''p''₁ ≤ ''p'' with ''p''_''i'' ∈ ''D''_''i''. Then ''G'' = is a ''D''-generic filter. The Rasiowa–Sikorski lemma can be viewed as an equivalent to a weaker form of

Martin's axiom In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consi ...

. More specifically, it is equivalent to MA(ℵ₀) and to the

axiom of countable choice The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function A with domain \mathbb ( ...

Examples

* For (''P'', ≤) = (Func(''X'', ''Y''), ⊇), the poset of

partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...

s from ''X'' to ''Y'', reverse-ordered by inclusion, define ''D''_''x'' = . Let ''D'' = . If ''X'' is countable, the Rasiowa–Sikorski lemma yields a ''D''-generic filter ''F'' and thus a function ''F'': ''X'' → ''Y''. * If we adhere to the notation used in dealing with ''D''-

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, suc ...

s, forms an ''H''-

. * If ''D'' is uncountable, but of

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

strictly smaller than 2^ℵ₀ and the poset has the

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. Overview There are really two conditions: the ''upwards'' and ''downwards'' countable c ...

, we can instead use

. However, Martin's axiom is not provable in ZFC.

References

* *

External links

* Timothy Chow's pape
A beginner's guide to forcing
is a good introduction to the concepts and ideas behind forcing. {{DEFAULTSORT:Rasiowa-Sikorski lemma Forcing (mathematics) Lemmas in set theory