Van Cittert–Zernike theorem: Difference between revisions
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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 (mathematics)|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 (mathematics)|filter]] ''F'' in ''P'' is called ''D''-[[generic filter|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 order|dense]] subsets of ''P'' then there exists a ''D''-generic [[filter (mathematics)|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''<sub>1</sub>, ''D''<sub>2</sub>, …. By assumption, there exists ''p'' ∈ ''P''. Then by density, there exists ''p''<sub>1</sub> ≤ ''p'' with ''p''<sub>1</sub> ∈ ''D''<sub>1</sub>. Repeating, one gets … ≤ ''p''<sub>2</sub> ≤ ''p''<sub>1</sub> ≤ ''p'' with ''p''<sub>''i''</sub> ∈ ''D''<sub>''i''</sub>. Then ''G'' = { ''q'' ∈ ''P'': ∃ ''i'', ''q'' ≥ ''p''<sub>''i''</sub>} 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(<math>\aleph_0</math>). | |||
== Examples == | |||
*For (''P'', ≥) = (Func(''X'', ''Y''), ⊂), the poset of [[partial function]]s from ''X'' to ''Y'', define ''D''<sub>''x''</sub> = {''s'' ∈ ''P'': ''x'' ∈ dom(''s'')}. If ''X'' is countable, the Rasiowa–Sikorski lemma yields a {''D''<sub>''x''</sub>: ''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]]s, {''H'' ∪ ''G''<sub>0</sub>: ''P''<sub>''ij''</sub>''P''<sub>''t''</sub>} forms an ''H''-[[generic filter]]. | |||
*If ''D'' is uncountable, but of [[cardinality]] strictly smaller than <math>2^{\aleph_0}</math> and the poset has the [[countable chain condition]], we can instead use [[Martin's axiom]]. | |||
== See also == | |||
*[[Generic filter]] | |||
*[[Martin's axiom]] | |||
== References == | |||
* ''Set Theory for the Working Mathematician''. Ciesielski, Krzysztof. Cambridge University Press, 1997. ISBN 0-521-59465-0 | |||
* {{cite book|first=Kenneth|last=Kunen|authorlink=Kenneth Kunen|title=[[Set Theory: An Introduction to Independence Proofs]]|publisher=North-Holland|year=1980|isbn=0-444-85401-0}} | |||
== 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 | |||
{{DEFAULTSORT:Rasiowa-Sikorski lemma}} | |||
[[Category:Forcing (mathematics)]] | |||
[[Category:Lemmas]] | |||
Revision as of 22:59, 17 September 2013
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 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 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 pi ∈ Di. Then G = { q ∈ P: ∃ i, q ≥ pi} 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().
Examples
- For (P, ≥) = (Func(X, Y), ⊂), the poset of partial functions from X to Y, define Dx = {s ∈ P: x ∈ dom(s)}. If X is countable, the Rasiowa–Sikorski lemma yields a {Dx: 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: PijPt} 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
- Set Theory for the Working Mathematician. Ciesielski, Krzysztof. Cambridge University Press, 1997. ISBN 0-521-59465-0
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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