Θ (set theory)

In set theory, Θ (pronounced like the letter theta) is the least nonzero 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 $(2^{{\aleph _{0}}})^{+}$, 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.

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 injection from the 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 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.

More easily, it can be shown that Θ is ${\displaystyle h(\mathbb {R} )}$ by definition, where $h$ is Hartogs' function, which maps an arbitrary set to the least well-ordered cardinal not less than or equal to it.