All Questions
Tagged with computability-theory ordinal-numbers
38 questions
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123
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Is possibile to define transfinite sum and product recursively? [closed]
On mathstackexchange a few days ago I published the following question where I asked about "transfinite" sum and products but actually nobody answered or gave an opinion with a comment: thus ...
0
votes
1
answer
137
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Is there a canonical mapping between countable transfinite ordinals and $\omega$? What about recursive ordinals?
Consider $\omega^2$. We can build a simple bijection between the ordinal and $\omega$ similarly to how the bijection between $\mathbb{Q}$ and $\mathbb{N}$ can be built.
I was wondering if there is a ...
- 103
5
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1
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487
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How to solve this exercise about large countable ordinals?
In this note (Notes on Higher Type ITTM-recursion, 2021) written by Philip Welch, I'm trying to solve exercise 3.5(i), but I don't know how to solve it.
The problem is: assume that $L_{\gamma_0}<_{...
- 1,490
6
votes
1
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571
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Parameter-free effective cardinals
In the paper "Effective cardinals and determinacy in third order arithmetic" by Juan Aguilera, effective cardinals is defined.
I'm curious about its little variation, parameter-free ...
- 1,490
7
votes
1
answer
443
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Gaps in the ordinals writable by Ordinal Turing Machines with a single countable parameter
Let $W(\alpha)$ denote the set of all (countable) ordinals writable by Ordinal Turing Machines with a single (countable) parameter $\alpha$, i.e. each computation starts with a single ($\alpha$-th) ...
- 959
3
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0
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368
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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 ...
- 1,318
2
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0
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235
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The supremum of ordinals eventually writable by Ordinal Turing Machines with an oracle for the class of stabilization ordinals
This question is based on the assumption that all computations start with no ordinal parameters (i.e. the input is empty).
The term “stabilization time of a machine” for this question implies the ...
- 959
4
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1
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268
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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 ...
- 959
4
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0
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253
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Is this recursion theoretic analogue of a criterion of weakly compact cardinal accurate?
Jensen proved that, if V=L, and $\kappa$ is a regular cardinal, then if for any stationary $A\subseteq \kappa$, the set $\{\alpha\mid A \text{ is stationary below }\alpha\}$ is stationary in $\kappa$, ...
- 1,490
2
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1
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204
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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? ...
- 1,490
4
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1
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469
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How large are the stabilization times of Ordinal Turing Machines with an oracle for the transfinite initial ordinals?
This question is based on the assumption that $V \ne L$ and we have $\omega_1^L < \omega_1$ (here $\omega_1^L$ is equal to the supremum of ordinals accidentally writable by no-oracle Ordinal Turing ...
- 959
4
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3
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403
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Countably infinite sets of ordinals as parameters for Ordinal Turing Machines
Let $A$ and $B$ denote two countably infinite sets of ordinals.
Let $W_A$ denote the supremum of ordinals writable by Ordinal Turing Machines with the set $A$ given as the source of parameters. That ...
- 959
4
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1
answer
227
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Is there a real $x$ which is eventually writable from an ordinal parameter $\alpha < \omega_1$, but not from $\omega_1$?
According to Lemma 3.14 in the paper “Recognizable sets and Woodin cardinals: Computation beyond the constructible universe”, there is a real $x$ in $L$ which is recognizable from some ordinal $\alpha$...
- 959
4
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1
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337
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How to compare three supremums of ordinals eventually writable by Ordinal Turing Machines?
This question implies that we have fixed: (i) a particular enumeration of Ordinal Turing machines; (ii) a particular way to encode an ordinal by an infinite binary sequence.
The class of $[1]$-...
- 959
4
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3
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495
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How large is the supremum of halting times of Infinite Time Turing Machines, assuming that halting times are bounded and inputs are arbitrary?
Given a fixed enumeration of Infinite Time Turing Machines (ITTMs), let $M_i(x)$ denote a computation of an $i$-th ITTM, assuming that the input $x$ is a real (an infinite binary sequence).
Then the ...
- 959
4
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3
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491
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Ordinal numbers reachable by primitive recursive ordinal functions in omega
$ \def \PRo {{\mathcal { PR } _ \omega}} $
The class of primitive recursive ordinal functions in the constant omega function (henceforth denoted by $ \PRo $) are defined by Jensen and Karp (1971) as ...
- 346
1
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1
answer
287
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How large is the smallest ordinal larger than any “minimal ordinal parameter” for any pair of an Ordinal Turing Machine and a real?
In this question, the notation $P^x(\alpha)$ denotes a situation where a particular OTM-program $P$ performs a computation on input $x$ with an ordinal parameter $\alpha$, assuming that $x$ is written ...
- 959
5
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0
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262
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Higher order arithmetic, hierarchies and proof theoretic ordinals
I asked this question on MSE some days ago but I have not received any answer so I have decided to post it here.
I would like to consider a generalization of the notation $\Pi$ and $\Sigma$ used for ...
- 161
9
votes
1
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610
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Does every cofinal branch through Kleene's O compute true arithmetic?
My question concerns cofinal branches through Kleene's $O$, which is a set of natural numbers and a computably enumerable relation $<_O$ on this set that provides
ordinal denotations for any ...
- 236k
10
votes
2
answers
595
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Transfinite algorithms
The Ford-Fulkerson algorithm is a classic algorithm that computes the maximum flow in a network. It is well-known that if irrational arc capacities are allowed, the algorithm does not necessarily ...
- 32.1k
5
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1
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193
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How far does this restricted definition on $\mathcal{O}$ goes?
$\mathcal{O}$ notation describes an onto function $f:\mathcal{O} \rightarrow \omega_{CK}$. In calculating all values $n \in \mathbb{N}$ such that $f(n)=\alpha$, when $\alpha$ is a limit, all indexes $...
- 881
5
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1
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749
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Which ordinal is larger, the supremum of ordinals writable by iterated Infinite Time Turing Machines or the smallest $\Sigma_2^1$-reflecting ordinal?
The ordinal $\tau_1$ corresponds to $\lambda^{\textit{it}}$ (the supremum of all ordinals writable by iterated ITTMs) — see Definition 3.1 in the paper “ITTMs with Feedback” [Robert S. Lubarsky]. ...
- 959
6
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0
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303
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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 ...
- 365
9
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1
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711
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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 ...
- 375
8
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0
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287
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Natural examples of recursive pseudowellorderings
Question: What are some natural examples of recursive pseudowellorderings?
By natural, I mean in the style of reasonable ordinal notation systems as opposed to dependent on a Gödel numbering or an ...
- 7,545
4
votes
2
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591
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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 ...
- 6,353
4
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0
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199
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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 $(...
- 32.5k
4
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2
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489
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Mapping between Notations
$\DeclareMathOperator{\address}{address}$
As in my other question, it is assumed that the (total) function describing a given notation is denoted as $\address:p \rightarrow \Bbb{N}$ and assumed to be ...
- 881
13
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1
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650
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About primitively recursively recognizable ordinals
Preliminary: I believe the notion of primitive recursive functions on ordinals is standard and unproblematic (the main difference with the finite case is that one needs to introduce a $\sup$ or $\...
- 32.5k
4
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1
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218
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Are two recursive well-orderings with the same order type recursively isomorphic?
Let $\leq_1$ and $\leq_2$ be recursive well-orderings of $\omega$ that have the same order type. Is there necessarily a recursive bijection $f$ such that $f(x)\leq_1f(y)\iff x\leq_2y$? If this does ...
- 2,130
9
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1
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442
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First order axioms for primitive recursion in Takeuti's theory of ordinal numbers
In this article, Takeuti has introduced a theory of ordinal numbers, which in his own words, is intended to be a first order theory:
The theory of ordinal numbers we are to develop is based on the ...
- 346
5
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1
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627
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On a characterization of the recursively inaccessible ordinals
For a given set of numbers $A$, let $O^A$ be the hyperjump of $A$. It is possible to iterate inductively the hyperjump of a set, through the computable ordinals, in a way that the $\alpha$-th ...
- 764
16
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1
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750
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Is ordinal arithmetic more complicated than classical arithmetic?
Consider the first-order language $\mathcal{L}_{\text{OA}}:=(+,\cdot,0,1)$; in this language, we can formulate statements of ordinal arithmetic. Clearly, the theory $T_{\text{OA}}$ of $(\text{On},+,\...
- 365
4
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0
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210
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Upper bound on ranks of well-founded trees in $SKI\Omega$ calculus
All ideas explained below are due to A.P.Goucher, and defined here.
First of all, $SKI\Omega$ calculus is an extension of standard SKI calculus, with additional type of combinator, called oracle ...
- 28.2k
12
votes
1
answer
834
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Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$?
Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, ...
- 6,819
27
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1
answer
2k
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Why isn't this a computable description of the ordinal of ZF?
In a previous MO question, I was told by several commenters that
(a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...
- 9,823
17
votes
1
answer
1k
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Looking for a copy of Leo Harrington's unpublished notes on the first nonprojectible ordinal
Sometime around 1975, Leo Harrington wrote a set of notes, apparently 13 pages long, entitled Kolmogorov's $R$-operator and the first nonprojectible ordinal. I do not know how widely they were ...
- 32.5k
17
votes
7
answers
2k
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Finding the largest integer describable with a string of symbols of predefined length
(This question is motivated by the reading of the article Large numbers and unprovable theorems by Joel Spencer, which can be found at http://mathdl.maa.org/images/upload_library/22/Ford/Spencer669-...
- 2,992