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| | In [[set theory]], a branch of mathematics, the '''condensation lemma''' is a result about sets in the |
| | [[constructible universe]]. |
| | |
| | It states that if ''X'' is a [[transitive set]] and is an [[elementary submodel]] of some level of the constructible hierarchy L<sub>α</sub>, that is, <math>(X,\in)\prec (L_\alpha,\in)</math>, then in fact there is some ordinal <math>\beta\leq\alpha</math> such that <math>X=L_\beta</math>. |
| | |
| | More can be said: If ''X'' is not transitive, then its [[transitive collapse]] is equal to some <math>L_\beta</math>, and the hypothesis of elementarity can be weakened to elementarity only for formulas which are <math>\Sigma_1</math> in the [[Lévy hierarchy]]. |
| | |
| | The lemma was formulated and proved by [[Kurt Gödel]] in his proof that the axiom of constructibility implies [[Continuum hypothesis#The generalized continuum hypothesis|GCH]]. |
| | |
| | == References == |
| | *{{cite book |
| | | last=Devlin |
| | | first=Keith |
| | | authorlink=Keith Devlin |
| | | year = 1984 |
| | | title = Constructibility |
| | | publisher = Springer |
| | | id = ISBN 3-540-13258-9 |
| | }} (theorem 5.2 and lemma 5.10) |
| | |
| | [[Category:Set theory]] |
| | [[Category:Lemmas]] |
| | |
| | |
| | {{settheory-stub}} |
Revision as of 09:02, 1 December 2013
In set theory, a branch of mathematics, the condensation lemma is a result about sets in the
constructible universe.
It states that if X is a transitive set and is an elementary submodel of some level of the constructible hierarchy Lα, that is, , then in fact there is some ordinal such that .
More can be said: If X is not transitive, then its transitive collapse is equal to some , and the hypothesis of elementarity can be weakened to elementarity only for formulas which are in the Lévy hierarchy.
The lemma was formulated and proved by Kurt Gödel in his proof that the axiom of constructibility implies GCH.
References
- 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 (theorem 5.2 and lemma 5.10)
Template:Settheory-stub