Recursive ordinal

In mathematics, specifically set theory, an ordinal $\alpha$ is said to be recursive if there is a recursive well-ordering of a subset of the natural numbers having the order type $\alpha$ .

It is trivial to check that $\omega$ is recursive, the successor of a recursive ordinal is recursive, and the set of all recursive ordinals is closed downwards. The supremum of all recursive ordinals is called the Church-Kleene ordinal and denoted by $\omega^{CK}_1$ . Indeed, an ordinal is recursive if and only if it is smaller than $\omega^{CK}_1$ . Since there are only countably many recursive relations, there are also only countably many recursive ordinals. Thus, $\omega^{CK}_1$ is countable.

单词	Recursive ordinal
释义	Recursive ordinal 英语百科 Recursive ordinal In mathematics, specifically set theory, an ordinal $\alpha$ is said to be recursive if there is a recursive well-ordering of a subset of the natural numbers having the order type $\alpha$ . It is trivial to check that $\omega$ is recursive, the successor of a recursive ordinal is recursive, and the set of all recursive ordinals is closed downwards. The supremum of all recursive ordinals is called the Church-Kleene ordinal and denoted by $\omega^{CK}_1$ . Indeed, an ordinal is recursive if and only if it is smaller than $\omega^{CK}_1$ . Since there are only countably many recursive relations, there are also only countably many recursive ordinals. Thus, $\omega^{CK}_1$ is countable.
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