Questions tagged [ordinal-computability]

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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 ...
2
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0answers
54 views

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. ...
2
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2answers
244 views

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 ...
3
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0answers
37 views

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 ...
4
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0answers
122 views

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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1answer
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 $\...
2
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0answers
119 views

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 ...
17
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2answers
2k views

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 ...
3
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1answer
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 ...