Definability and decision problems in arithmetic

Robinson, Julia

doi:10.2307/2266510

Cited by 270 publications

(149 citation statements)

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“…His predicates, like ours, are based on Hasse's Norm Theorem, but he uses it differently than we do. The author also thanks Denef for pointing out the results of Julia Robinson in [8].…”

Section: Given a Global Field Kmentioning

confidence: 95%

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Undecidability and Definability for the Theory of Global Fields

Rumely¹

1980

Transactions of the American Mathematical Society

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Abstract.We prove that the theory of global fields is essentially undecidable, using predicates based on Hasse's Norm Theorem to define valuations. Polynomial rings or the natural numbers are uniformly defined in all global fields, as well as Godel functions encoding finite sequences of elements.We will prove that the elementary theory of global fields is essentially undecidable. While this is a negative logical result, our proofs have the positive consequence that a great variety of number-theoretic objects, from rings of integers and valuations, to zeta-functions and adele rings, can be discussed in the theory of global fields. It is our hope that the theory may eventually be a vehicle for applying logical methods in number theory.Our main theorems are as follows.

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“…His predicates, like ours, are based on Hasse's Norm Theorem, but he uses it differently than we do. The author also thanks Denef for pointing out the results of Julia Robinson in [8].…”

Section: Given a Global Field Kmentioning

confidence: 95%

“…However, in any case Julia Robinson has shown [8] that there is a uniform way of defining the natural numbers in the ring of integers of a number field. Her results may be formulated as follows.…”

Section: Locallymentioning

confidence: 99%

Undecidability and Definability for the Theory of Global Fields

Rumely¹

1980

Transactions of the American Mathematical Society

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“…Let W be a recursive set of rational primes. Let [19], [20] and [24] for more details.) Unfortunately, we have been unsuccessful in obtaining the analogous definability results for infinite W .…”

Section: Introductionmentioning

confidence: 99%

On Diophantine definability and decidability in some infinite totally real extensions of ℚ

Shlapentokh

2003

Trans. Amer. Math. Soc.

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Abstract. Let M be a number field, and W M a set of its non-Archimedean. . , pr} be a finite set of prime numbers. Let F inf be the field generated by all the p j i -th roots of unity as j → ∞ and i = 1, . . . , r. Let K inf be the largest totally real subfield of F inf . Then for any ε > 0, there exist a number field M ⊂ K inf , and a set W M of non-Archimedean primes of M such that W M has density greater than 1 − ε, and Z has a Diophantine definition over the integral closure of O M,W M in K inf .

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“…Church [7], and the essential undecidability (undecidability of its every consistent extension) by B. Rosser [15], both as early as 1936. Consequences of this result are the undecidability of the theory of natural numbers with multiplication and successor function and with divisibility and successor function, both discovered by J. Robinson in [14]. To complete the picture, the existential fragment of the full arithmetic i.e., Hilbert's Tenth Problem was proved undecidable by Y. Matiyasevich [12].…”

Section: Introductionmentioning

confidence: 99%

On Decidability Within the Arithmetic of Addition and Divisibility

Bozga

Iosif

2005

Foundations of Software Science and Computational Structures

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Abstract. The arithmetic of natural numbers with addition and divisibility has been shown undecidable as a consequence of the fact that multiplication of natural numbers can be interpreted into this theory, as shown by J. Robinson [14]. The most important decidable subsets of the arithmetic of addition and divisibility are the arithmetic of addition, proved by M. Presburger [13], and the purely existential subset, proved by L. Lipshitz [11]. In this paper we define a new decidable fragment of the form QzQ1x1 . . . Qnxnϕ(x, z) where the only variable allowed to occur to the left of the divisibility sign is z. For this form, called L| in the paper, we show the existence of a quantifier elimination procedure which always leads to formulas of Presburger arithmetic. We generalize the L IntroductionThe undecidability of first-order arithmetic of natural numbers occurs as a consequence of Gödel's Incompleteness Theorem [10]. The basic result has been discovered by A. Church [7], and the essential undecidability (undecidability of its every consistent extension) by B.

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Definability and decision problems in arithmetic

Cited by 270 publications

References 5 publications

Undecidability and Definability for the Theory of Global Fields

Undecidability and Definability for the Theory of Global Fields

On Diophantine definability and decidability in some infinite totally real extensions of ℚ

On Decidability Within the Arithmetic of Addition and Divisibility

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