Theta (set theory)

In set theory, Θ is the least 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 , the cardinal successor of the cardinality of the continuum. However, Θ is often studied in contexts where the axiom of choice fails, such as  models of the axiom of determinacy.Θ is also the  supremum of the lengths of all prewellorderings of the reals.
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