Questions tagged [ordinal-computability]
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20 questions
6 votes
2 answers
370 views
Extending polynomial hierarchy above $\omega$
The arithmetic hierarchy is naturally extended to all ordinals via ordinal notations creating a hierarchy for all hyperarithmetic sets. The polynomial time hierarchy is defined analogously to the ...
- 3,977
9 votes
0 answers
217 views
Logical properties of realizability (topoi or McCarty models) defined by alpha-recursion on admissible ordinals
Setup: Let $\alpha$ be an admissible ordinal (viꝫ., one such that $L_\alpha$ is a model of Kripke-Platek set theory), identified as usual with the set of ordinals $<\alpha$. Then there is a ...
- 38k
3 votes
1 answer
617 views
Are all constructible from below sets parameter free definable?
Lets take the intersection of the theory of $L_{\omega_1^{CK}}$ and $\sf ZF + [V=L]$, this is equivalent to the theory of constructability from below + limit stages. Can this theory prove the ...
- 13k
1 vote
0 answers
54 views
Single parameter programs and halting supremums
Let's first define a function $H:\mathrm{Ord} \rightarrow \mathrm{Ord}$. Suppose we are given some ordinal $\alpha$ and we want to find $H(\alpha)$. We define $H(\alpha)$ as the supremum of halting ...
- 901
3 votes
0 answers
441 views
An alternative definition of computable ordinals
An ordinal $\alpha$ is said to be computable if there is a computable relation on a subset of integers that is well-ordered and its order type equals $\alpha$. But let's consider well-founded trees on ...
- 1,328
4 votes
1 answer
306 views
Existence of a particular function that maps an arbitrary set of ordinals to a single ordinal
Does there exist a function $f$ that satisfies all of the following three properties? The function converts an arbitrarily large (empty, finite, countably/uncountably infinite) set of ordinals to a ...
- 1,029
0 votes
1 answer
323 views
Is there a non-standard model of PA computable with infinitary computation?
By the Tennenbaum's theorem, there are no non-standard countable models of Peano Arithmetic that are computable using Turing machines. What about models of infinitary computation like infinite time ...
- 127
2 votes
1 answer
256 views
Ordinal notations in α-recursion theory
Is there a theory about using α-recursion to compute ordinals? For example, consider α-recursive well orders on α, what is the supreme of their order type? Is it the next admissible ordinal after α? ...
- 1,652
1 vote
2 answers
929 views
Is there an analogue of the Lost Melody Theorem in ordinary recursion theory and if not, why not?
In their arXiv preprint, "Infinite Time Turing Machines" (arXiv:math/9808093v1 [math.LO] 21 Aug 1998) Hamkins and Lewis state the Lost Melody Theorem for ITTM's as follows: Lost Melody ...
- 6,260
5 votes
1 answer
299 views
Can there be a computational characterization of HOD?
This is a somewhat open-ended question, so let me motivate it a little. One of the first results in the study of Ordinal Turing Machines includes the following computational characterization of $L$ by ...
- 1,305
7 votes
0 answers
364 views
Weaker versions of Gandy ordinals
Gostanian's paper "The next admissible ordinal" (see https://www.sciencedirect.com/science/article/pii/0003484379900251 ), is concerned with the supremum of the $\alpha$-recursive ordinals for various ...
- 375
10 votes
1 answer
795 views
Computable models of the ordinal numbers
It's known, for example in the answer to this question: Is there a computable model of ZFC? that ZFC has no computable model. My questions is: is there a model of ZFC for which the order relation on ...
- 385
2 votes
0 answers
83 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. ...
- 901
4 votes
2 answers
697 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 ...
- 7,130
3 votes
0 answers
89 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 ...
- 901
4 votes
0 answers
234 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 $(f_e)...
- 38k
8 votes
2 answers
840 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 $\...
- 2,281
2 votes
0 answers
160 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 ...
- 6,260
18 votes
2 answers
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 ...
- 18.3k
4 votes
1 answer
486 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 ...
- 6,260