An Undecidable Linear Order That Is $n$-Decidable for All $n$

Chisholm, John; Moses, Michael

doi:10.1305/ndjfl/1039118866

Cited by 6 publications

(5 citation statements)

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“…2. Using Proposition 5.5, we may obtain the result of Chisholm and Moses [6] saying that there is a linear ordering 38 such that D n (38) is computable for all n, but D c (33) is not computable. We choose an w-table (C")" 6w such that C" is computable for all n, but ©"C" is not computable.…”

Section: C (^)= T D 2n -\(^)-mentioning

confidence: 99%

Sequences of n-diagrams

2002

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We consider only computable languages, and countable structures, with universe a subset of ω, which we think of as a set of constants. We identify sentences with their Gödel numbers. Thus, for a structure , the complete (elementary) diagram, Dc(), and the atomic diagram, D(), are subsets of ω. We classify formulas as usual. A formula is both Σ0 and Π0 if it is open. For n > 0, a formula, in prenex normal form, is Σn, or Πn, if it has n blocks of like quantifiers, beginning with ∃, or ∀. For a formula θ, in prenex normal form, we let neg(θ) denote the dual formula that is logically equivalent to ¬θ—if θ is Σn, then neg(θ) is Πn, and vice versa.

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Section: C (^)= T D 2n -\(^)-mentioning

confidence: 99%

Sequences of n-diagrams

2002

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“…Moses [67] showed that for every n ≥ 1, there is a linear order that is n-decidable, but has no (n + 1)-decidable copy. Chisholm and Moses [12] have shown that there is a linear order that is n-decidable for every n ∈ , but has no decidable copy. Goncharov [22] established similar results for Boolean algebras.…”

Section: Effective Completeness Theorem a Decidable Theory Has A Decmentioning

confidence: 99%

“…Theorem 5.5 (Chisholm and Moses [12]). There is a structure A that is ndecidable and whose every computable copy is n-decidable, for all n, but A has no decidable copy.…”

Section: (C 1 )mentioning

confidence: 99%

Computability-Theoretic Complexity of Countable Structures

Harizanov¹

2002

Bull. symb. log.

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Computable model theory, also called effective or recursive model theory, studies algorithmic properties of mathematical structures, their relations, and isomorphisms. These properties can be described syntactically or semantically. One of the major tasks of computable model theory is to obtain, whenever possible, computability-theoretic versions of various classical model-theoretic notions and results. For example, in the 1950's, Fröhlich and Shepherdson realized that the concept of a computable function can make van der Waerden's intuitive notion of an explicit field precise. This led to the notion of a computable structure. In 1960, Rabin proved that every computable field has a computable algebraic closure. However, not every classical result “effectivizes”. Unlike Vaught's theorem that no complete theory has exactly two nonisomorphic countable models, Millar's and Kudaibergenov's result establishes that there is a complete decidable theory that has exactly two nonisomorphic countable models with computable elementary diagrams. In the 1970's, Metakides and Nerode [58], [59] and Remmel [71], [72], [73] used more advanced methods of computability theory to investigate algorithmic properties of fields, vector spaces, and other mathematical structures.

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“…In particular, such a model is said to be N -decidable if the set of the Σ N -sentences of its elementary diagram is computable. A seminal result in this direction is the theorem of Moses and Chisholm that there is a computable linear order that is n-decidable for all n yet not decidable [4]. More recently, Fokina et.…”

Section: Introductionmentioning

confidence: 99%

On the complexity of the theory of a computably presented metric structure

Camrud¹,

Goldbring²,

McNicholl³

2021

Preprint

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We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form φ M ≤ r, and the open diagram, which encapsulates strict inequalities of the form φ M < r. We show that the closed and open Σ N diagrams are Π 0 N+1 and Σ N respectively, and that the closed and open Π N diagrams are Π 0 N and Σ 0 N+1 respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal.

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An Undecidable Linear Order That Is $n$-Decidable for All $n$

Cited by 6 publications

References 4 publications

Sequences of n-diagrams

Sequences of n-diagrams

Computability-Theoretic Complexity of Countable Structures

On the complexity of the theory of a computably presented metric structure

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