# Questions tagged [ordinal-computability]

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9
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### Question about relation of admissibility and cofinality (of infinite-time functions)

Due to its nature, this is a very long question (so please bear with me). To convey the question, while keeping it in manageable length, I have tried to convey the main idea in the question (rather ...

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### Length of Gaps in Clockable Values

As I understand, there can't be any (total) OTM computable function (in traditional input/output sense) $f:Ord \rightarrow Ord$ such that $f(x)$ gives an upper-bound on the length of the $x$-th gap.
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### Connection between countable ordinals and Turing degrees

$\omega^{CK}_1$ is the supremum of all the recursive ordinals, where an ordinal $\alpha$ is recursive if there is a computable ordering of a subset of the naturals with order type $\alpha$.
For a ...

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### Trace-Recursive Functions and Natural/Unnatural Operations

I have been quite hesitant to post this question. Due to the highly general nature of the question there is a possibility of a trivial answer. At a first glance at least, one gets the feeling that ...

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### On the proof of a normal form theorem for ordinal (primitive) recursion

Consider the following statement (which follows easily from various results found in the literature):
(†) There exists a primitive recursive (“p.r.”) relation $T$ on the ordinals such that, if $(...

**7**

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227 views

### Order type of $\alpha$-computable well-orderings

One of the nice features of the first admissible ordinal after $\omega$, i.e. $\omega_1^{CK}$, is that it is the collection of ordinals whose order type is that of a computable well-ordering on $\...

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### A question regarding an analogue of the Kleene $T$-predicate for Koepke's ordinal computability

Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence ...

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### Analogues of Primitive Recursive Functions

Let $\mathbf{A}$ be an admissible set (possibly with urelements). I am wondering if there is some good notion of "primitive recursive arithmetic" relative to $\mathbf{A}$. More precisely, I would like ...

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310 views

### At what level of the analytic hierarchy do Cohen reals lie?

In his doctoral thesis titled "Three models of ordinal computability", Benjamin Seyfferth proved the following theorems:
i) A set $\mathtt A$ of reals is Ordinal Turing Machine-enumerable if and only ...