Decidability and Classification of the Theory of Integers With Primes

Kaplan, Itay; Shelah, Saharon

doi:10.1017/jsl.2017.16

Cited by 11 publications

(13 citation statements)

References 16 publications

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“…We will next prove that Th(Z, SF Z ) is k-independent for all k > 0; see [5] for a definition of k-independence. The proof is almost the exact replica of the proof in [9] except the necessary modifications taken in the current paragraph. Suppose l > 0, and S is an arbitrary subset of {0, ... , l -1}.…”

Section: Lemma 33 If ϕ(X Z) Is a P-condition Then Modulo Either Sf *mentioning

confidence: 81%

“…Introduction. In [9], Kaplan and Shelah showed under the assumption of Dickson's conjecture that if Z is the additive group of integers implicitly assumed to contain 1 as a distinguished constant and a →a as a distinguished function, and if Pr is the set of a ∈ Z such that either a ora is prime, then the theory of (Z; Pr) is model complete, decidable, and super-simple of U-rank 1. From our current point of view, the above result can be seen as an example of a more general phenomenon where we can often capture aspects of randomness inside a structure using firstorder logic and deduce in consequence several model-theoretic properties of that structure.…”

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confidence: 99%

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The Additive Groups of and With Predicates for Being Square-Free

Bhardwaj¹,

Tran²

2020

J. symb. log.

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We consider the structures $(\mathbb {Z}; \mathrm {SF}^{\mathbb {Z}})$ , $(\mathbb {Z}; <, \mathrm {SF}^{\mathbb {Z}})$ , $(\mathbb {Q}; \mathrm {SF}^{\mathbb {Q}})$ , and $(\mathbb {Q}; <, \mathrm {SF}^{\mathbb {Q}})$ where $\mathbb {Z}$ is the additive group of integers, $\mathrm {SF}^{\mathbb {Z}}$ is the set of $a \in \mathbb {Z}$ such that $v_{p}(a) < 2$ for every prime p and corresponding p-adic valuation $v_{p}$ , $\mathbb {Q}$ and $\mathrm {SF}^{\mathbb {Q}}$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

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Section: Lemma 33 If ϕ(X Z) Is a P-condition Then Modulo Either Sf *mentioning

confidence: 81%

mentioning

confidence: 99%

The Additive Groups of and With Predicates for Being Square-Free

Bhardwaj¹,

Tran²

2020

J. symb. log.

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“…Concerning reducts of Presburger arithmetic, a recent result of the first author [6] is that there are no structures strictly between (Z, +, 0) and (Z, +, <, 0). In a different direction, Kaplan and Shelah [11] show that if P = {z ∈ Z : |z| is prime} then (Z, +, 0, P ) is unstable and, assuming a fairly strong conjecture in number theory, (Z, +, 0, P ) is supersimple of SU -rank 1 (see also Remark 1.8(3) below).…”

Section: Introduction and Summary Of Main Resultsmentioning

confidence: 99%

Stable groups and expansions of $(\mathbb Z,+,0)$

Conant

Pillay

2018

Fund. Math.

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We show that if G is a sufficiently saturated stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then G is superstable of finite U -rank. Combined with recent work of Palacín and Sklinos, we conclude that (Z, +, 0) has no proper stable expansions of finite weight. A corollary of this result is that if P ⊆ Z n is definable in a finite dp-rank expansion of (Z, +, 0), and (Z, +, 0, P ) is stable, then P is definable in (Z, +, 0). In particular, this answers a question of Marker on stable expansions of the group of integers by sets definable in Presburger arithmetic.Theorem 1.2. If P ⊆ Z n is definable in a finite dp-rank expansion of (Z, +, 0), and (Z, +, 0, P ) is stable, then P is definable in (Z, +, 0).The notion of dp-rank in NIP theories has been an important tool in extending the work of stability theory to the unstable setting (see, e.g., [23]), and so Theorem 1.2 establishes a fundamental fact about the behavior of NIP expansions of (Z, +, 0). The proof of this theorem will be obtained from a more general result on stable

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“…While the ring structure of the integers is model-theoretically extremely wild for being subject to the Gödel phenonemon, tame reducts of this structure with traces of multiplication, have been subject of various literature, see for example [2,4,5,6,7]. A classical result in this direction, is the tameness of the structure (Z, nZ, +, −, <), the so called Pressburger arithematic.…”

Section: Introductionmentioning

confidence: 99%

The Additive Structure of Integers with the Lower Wythoff Sequence

Khani,

Zarei

2019

Preprint

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Decidability and Classification of the Theory of Integers With Primes

Cited by 11 publications

References 16 publications

The Additive Groups of and With Predicates for Being Square-Free

The Additive Groups of and With Predicates for Being Square-Free

Stable groups and expansions of $(\mathbb Z,+,0)$

The Additive Structure of Integers with the Lower Wythoff Sequence

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