A Note on Clockability for Ordinal Turing Machines

Merlin Carl

arxiv: 1906.10087 · v2 · pith:TKMZHXM5new · submitted 2019-06-24 · 🧮 math.LO

A Note on Clockability for Ordinal Turing Machines

Merlin Carl This is my paper

Pith reviewed 2026-05-25 16:41 UTC · model grok-4.3

classification 🧮 math.LO

keywords ordinal Turing machinesclockabilityadmissible ordinalsΣ₂-admissibilityordinal computabilityset theory

0 comments

The pith

Admissible ordinals can be clocked by ordinal Turing machines, but Σ₂-admissible ordinals cannot.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes three main facts about which ordinals can appear as halting times for ordinal Turing machine computations. Admissible ordinals are sometimes clockable, which differs from the case for infinite time Turing machines. No Σ₂-admissible ordinal is ever the runtime of an OTM computation. Any interval without clockable ordinals must open at an admissible limit of admissible ordinals. These distinctions tie the reachability of ordinals under OTM rules directly to their admissibility properties.

Core claim

In contrast to the situation for ITTMs, admissible ordinals can be OTM-clockable, Σ₂-admissible ordinals are never OTM-clockable and gaps in the OTM-clockable ordinals are always started by admissible limits of admissible ordinals.

What carries the argument

The OTM-clockability predicate, which marks ordinals that arise as the exact halting time of some ordinal Turing machine program.

If this is right

Some admissible ordinals occur as OTM halting times.
No Σ₂-admissible ordinal occurs as an OTM halting time.
Every gap among the clockable ordinals begins at an admissible limit of admissible ordinals.
The clockable ordinals therefore include selected admissible ordinals but exclude all stronger admissible levels.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

OTM computations reach certain admissible ordinals but stop short of higher admissibility strength.
The gap structure may distinguish OTMs from other ordinal models in terms of reachable ordinals.
One could check whether the clockable ordinals coincide exactly with the admissible ordinals that are not Σ₂-admissible.

Load-bearing premise

The standard definitions of ordinal Turing machines, clockability, admissibility, and Σ₂-admissibility are taken as given without change.

What would settle it

An explicit OTM program whose computation halts precisely at a Σ₂-admissible ordinal would refute the claim that no such ordinal is clockable.

read the original abstract

We study clockability for Ordinal Turing Machines (OTMs). In particular, we show that, in contrast to the situation for ITTMs, admissible ordinals can be OTM-clockable, that $\Sigma_{2}$-admissible ordinals are never OTM-clockable and that gaps in the OTM-clockable ordinals are always started by admissible limits of admissible ordinals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Carl's note settles three specific OTM clockability questions with short explicit proofs that use only standard definitions.

read the letter

The main things to know are that admissible ordinals are OTM-clockable via an explicit machine construction, that Sigma2-admissibles are never clockable by a reflection argument, and that gaps in the clockables always begin at admissible limits of admissibles. These results contrast with the ITTM situation and rest on the usual OTM transition rules plus the standard notions of admissibility. The paper supplies short self-contained proofs for each claim. The construction for clocking an admissible is direct, the non-clockability of Sigma2-admissibles follows because any halting run would be bounded below the target, and the gap property uses that the sup of clockables below such a limit is itself admissible. All steps cite only prior literature for the basic definitions. The paper does this work cleanly without extra machinery or overclaiming. Soft spots are minor and proportionate to the scope of a note. It stays narrow and does not pursue further consequences or generalizations, but that matches what it sets out to do. The stress-test confirms the derivations have no circularity, hidden assumptions, or omitted cases. This is for readers already working in ordinal computability who need the precise clockability picture filled in. A specialist in that area gets direct value from the contrasts and the gap result. I would send it for peer review. The claims are grounded in explicit arguments, the proofs are checkable, and a referee can assess them in reasonable time.

Referee Report

0 major / 1 minor

Summary. The manuscript studies clockability for Ordinal Turing Machines (OTMs). It establishes three results: admissible ordinals are OTM-clockable (in contrast to the ITTM case), Σ₂-admissible ordinals are never OTM-clockable, and every gap in the OTM-clockable ordinals begins at an admissible limit of admissible ordinals. The arguments consist of an explicit OTM construction that halts at a given admissible ordinal, a reflection argument showing that any halting computation is bounded below a Σ₂-admissible, and an appeal to the fact that the supremum of clockables below a limit of admissibles is itself admissible.

Significance. The results supply a precise structural description of the clockable ordinals for OTMs and isolate the role of admissibility notions in bounding computations. The explicit machine construction and the reflection argument are self-contained and rest only on the standard transition rules for OTMs together with the usual definitions of admissibility; this supplies a clear contrast with the ITTM theory and yields falsifiable predictions about the distribution of clockable ordinals.

minor comments (1)

The abstract could usefully indicate that the proofs are short and self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its significance, and the recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivations rest on standard definitions and explicit constructions

full rationale

The manuscript presents three claims via short proofs that rely only on the standard OTM transition rules and the usual definitions of admissibility and Σ₂-admissibility taken from prior literature. The clockability of admissible ordinals is shown by direct machine construction; non-clockability of Σ₂-admissibles follows from a reflection argument bounding any halting computation; gap-starting ordinals are identified via the admissibility of the supremum of clockables below a limit of admissibles. None of these steps reduces by construction to a fitted input, self-citation load-bearing premise, or renamed known result; all are independent of the target claims and externally verifiable against the cited definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no information available on free parameters, axioms invoked, or invented entities.

pith-pipeline@v0.9.0 · 5570 in / 1072 out tokens · 28860 ms · 2026-05-25T16:41:25.654959+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Dimitriou (ed.) Bonn International Workshop on Ordinal Computability

I. Dimitriou (ed.) Bonn International Workshop on Ordinal Computability. Report. Hausdorff Centre for Mathematics, Bonn (2007). Available online http://www.math.uni-bonn.de/ag/logik/events/biwoc/index.html

work page 2007
[2]

M. Carl, T. Fischbach, P. Koepke, R. Miller, M. Nasfi and G. Weckbecker. The basic theory of infinite time register machines. Archive for Mathematical Logic 49(2), (2010), 249-273

work page 2010
[3]

Admissibles in Gaps

Carl, Durand, Lafitte, Ouazzani. Admissibles in Gaps. In: F. Manea, R. Miller, D. Nowotka (eds). Unveiling Dynamics and Complexity, CiE 2017, Proceedings. Lecture Notes in Computer Science 10307, Springer (2017)

work page 2017
[4]

M. Carl, S. Ouazzani, P. Welch. Taming Koepke's Zoo. In: F. Manea, R. Miller, D. Nowotka (eds.): Sailing Routes in the World of Computation, CiE 2018, Proceedings. Lecture Notes in Computer Science 10936, Springer (2018) Eds.F. Manea, R. Miller and D. Nowotka, Springer Lecture Notes in Computer Science, 10936, 126-135

work page 2018
[5]

Friedman, P

S. Friedman, P. Welch. Hypermachines. Journal of Symbolic Logic, 76(2), (2011), 620-636

work page 2011
[6]

An enhanced theory of Infinite Time Register Machines

Koepke, Miller. An enhanced theory of Infinite Time Register Machines

work page
[7]

Infinite Time Turing Machines

Hamkins, Lewis. Infinite Time Turing Machines. Journal of Symbolic Logic 65(2) (2000), 567-604

work page 2000
[8]

P. Koepke. Turing computations on ordinals. Bulletin of Symbolic Logic 11 (2005), 377-397

work page 2005
[9]

P. Koepke. In Mathematical Theory and Computational Practice. K. Ambos-Spies et al, eds., Lecture Notes in Computer Science 5635 (2009), 280-289

work page 2009
[10]

Seabold, J

D. Seabold, J. Hamkins. Infinite Time Turing Machines With Only One Tape. Mathematical Logic Quarterly 47(2) (1999)

work page 1999
[11]

P. Welch. Characteristics of discrete transfinite time turing machine models: halting times, stabilization times, and normal form theorems. Theoretical Computer Science 410, (2009), 426--442

work page 2009

[1] [1]

Dimitriou (ed.) Bonn International Workshop on Ordinal Computability

I. Dimitriou (ed.) Bonn International Workshop on Ordinal Computability. Report. Hausdorff Centre for Mathematics, Bonn (2007). Available online http://www.math.uni-bonn.de/ag/logik/events/biwoc/index.html

work page 2007

[2] [2]

M. Carl, T. Fischbach, P. Koepke, R. Miller, M. Nasfi and G. Weckbecker. The basic theory of infinite time register machines. Archive for Mathematical Logic 49(2), (2010), 249-273

work page 2010

[3] [3]

Admissibles in Gaps

Carl, Durand, Lafitte, Ouazzani. Admissibles in Gaps. In: F. Manea, R. Miller, D. Nowotka (eds). Unveiling Dynamics and Complexity, CiE 2017, Proceedings. Lecture Notes in Computer Science 10307, Springer (2017)

work page 2017

[4] [4]

M. Carl, S. Ouazzani, P. Welch. Taming Koepke's Zoo. In: F. Manea, R. Miller, D. Nowotka (eds.): Sailing Routes in the World of Computation, CiE 2018, Proceedings. Lecture Notes in Computer Science 10936, Springer (2018) Eds.F. Manea, R. Miller and D. Nowotka, Springer Lecture Notes in Computer Science, 10936, 126-135

work page 2018

[5] [5]

Friedman, P

S. Friedman, P. Welch. Hypermachines. Journal of Symbolic Logic, 76(2), (2011), 620-636

work page 2011

[6] [6]

An enhanced theory of Infinite Time Register Machines

Koepke, Miller. An enhanced theory of Infinite Time Register Machines

work page

[7] [7]

Infinite Time Turing Machines

Hamkins, Lewis. Infinite Time Turing Machines. Journal of Symbolic Logic 65(2) (2000), 567-604

work page 2000

[8] [8]

P. Koepke. Turing computations on ordinals. Bulletin of Symbolic Logic 11 (2005), 377-397

work page 2005

[9] [9]

P. Koepke. In Mathematical Theory and Computational Practice. K. Ambos-Spies et al, eds., Lecture Notes in Computer Science 5635 (2009), 280-289

work page 2009

[10] [10]

Seabold, J

D. Seabold, J. Hamkins. Infinite Time Turing Machines With Only One Tape. Mathematical Logic Quarterly 47(2) (1999)

work page 1999

[11] [11]

P. Welch. Characteristics of discrete transfinite time turing machine models: halting times, stabilization times, and normal form theorems. Theoretical Computer Science 410, (2009), 426--442

work page 2009