Tennenbaum's theorem for models of arithmetic

Kaye, Robert

doi:10.1017/cbo9780511910616.005

Cited by 3 publications

(15 citation statements)

References 23 publications

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“…The proof in subsection 7.1 we will assume enumerability of the model, enabling a very direct diagonal argument [BBJ02]. In subsection 7.2 we look at the proof approach that is most prominently found in the literature [Smi14,Kay11] and uses the existence of recursively inseparable sets.…”

Section: Tennenbaum's Theoremmentioning

confidence: 99%

“…Via Inseparable Predicates. The most frequently reproduced proof of Tennenbaum's theorem [Kay11,Smi14] uses the existence of recursively inseparable sets and nonstandard coding to establish the existence of a non-recursive set.…”

Section: Tennenbaum's Theoremmentioning

confidence: 99%

“…A model is considered computable if its elements can be coded by numbers in N, and the arithmetic operations on model elements can be realized by computable functions on these codes. Usually, this theorem is formulated in a classical framework such as ZF set theory, and the precise meaning of computable is given by making reference to a concrete model of computation like Turing machines, µ-recursive functions, or the λ-calculus [Kay11,Smi14]. In a case like this, where computability theory is applied rather than developed, the computability of a function is rarely proven by exhibiting an explicit construction in the specific model, but rather by invoking the informal Church-Turing thesis, which states that every function intuitively computable will be computable in the chosen model.…”

Section: Introductionmentioning

confidence: 99%

“…In the sketched framework, we follow the classical presentations of Tennenbaum's theorem [Kay11,Smi14] to develop constructive versions that only assume a type-theoretic version of Markov's principle [MC17]. This is then complemented by the adaption of an inherently constructive variant given by McCarty [McC87,McC88].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Analysis of Tennenbaum's Theorem in Constructive Type Theory

Hermes¹,

Kirst²

2023

Preprint

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Tennenbaum's theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit, analyze and generalize this result.The chosen framework allows for a synthetic approach to computability theory, exploiting that, externally, all functions definable in constructive type theory can be shown computable. We then build on this viewpoint and furthermore internalize it by assuming a version of Church's thesis, which expresses that any function on natural numbers is representable by a formula in PA. This assumption provides for a conveniently abstract setup to carry out rigorous computability arguments, even in the theorem's mechanization.Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum's theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant.

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Section: Tennenbaum's Theoremmentioning

confidence: 99%

Section: Tennenbaum's Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An Analysis of Tennenbaum's Theorem in Constructive Type Theory

Hermes¹,

Kirst²

2023

Preprint

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“…To obtain categoricity, we then invoke Tennenbaum's Theorem [1959], that all computable models of the first-order Peano axioms for arithmetic are isomorphic (for proofs, see Kaye [2011]; Ash and Knight [2000, p. 59]). However, this option faces the obvious challenge of cultivating a notion of computability that is sufficiently independent from the arithmetical notions it seeks to vouchsafe.…”

Section: Model-theoretic Scepticism and Intermediary Logicsmentioning

confidence: 99%

Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics

Button¹,

Walsh²

2016

Philosophia Mathematica

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This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent 'internal' renditions of the famous categoricity arguments for arithmetic and set theory.

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An Analysis of Tennenbaum's Theorem in Constructive Type Theory

Hermes,

Kirst

2024

Logical Methods in Computer Science

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Tennenbaum's theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit, analyze and generalize this result. The chosen framework allows for a synthetic approach to computability theory, exploiting that, externally, all functions definable in constructive type theory can be shown computable. We then build on this viewpoint, and furthermore internalize it by assuming a version of Church's thesis, which expresses that any function on natural numbers is representable by a formula in PA. This assumption provides for a conveniently abstract setup to carry out rigorous computability arguments, even in the theorem's mechanization. Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum's theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant.

show abstract

Tennenbaum's theorem for models of arithmetic

Cited by 3 publications

References 23 publications

An Analysis of Tennenbaum's Theorem in Constructive Type Theory

An Analysis of Tennenbaum's Theorem in Constructive Type Theory

Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics

An Analysis of Tennenbaum's Theorem in Constructive Type Theory

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