en>David Eppstein: source

2012-08-15T17:23:27Z

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In [[set theory]], '''Θ''' (pronounced like the letter [[theta]]) is the least nonzero [[ordinal number|ordinal]] α such that there is no [[surjection]] from the reals onto α.

If the [[axiom of choice]] (AC) holds (or even if the reals can be [[wellordered]]) then Θ is simply <math>(2^{\aleph_0})^+</math>, the cardinal successor of the [[cardinality of the continuum]]. However, Θ is often studied in contexts where the axiom of choice fails, such as [[model theory|models]] of the [[axiom of determinacy]].

Θ is also the [[supremum]] of the lengths of all [[prewellordering]]s of the reals.

==Proof of existence==
It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto which there is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinals are wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the set of all prewellorderings of the reals having length α. This would give an [[injective function|injection]] from the [[class (set theory)|class]] of all ordinals into the set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the [[axiom of power set|powerset axiom]]). Now the [[axiom of replacement]] shows that the class of all ordinals is in fact a set. But that is impossible, by the [[Burali-Forti paradox]].

{{DEFAULTSORT:Theta (Set Theory)}}
[[Category:Cardinal numbers]]
[[Category:Descriptive set theory]]
[[Category:Determinacy]]

{{settheory-stub}}

Compression theorem - Revision history

en>David Eppstein: source