Solvability of the theory of integers with addition, order, and multiplication by an arbitrary number

Penzin, Yu. G.

doi:10.1007/bf01147467

Cited by 3 publications

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“…The same properties—bounded pp‐elimination , (consequently) model completeness and decidability (cf. and independently )—have been be shown also for

{LA}_{1}

. For

κ \geq 2

, nevertheless,

{LA}_{κ}

was only known to satisfy quantifier elimination up to bounded formulas .…”

Section: Introductionmentioning

confidence: 92%

A wild model of linear arithmetic and discretely ordered modules

Glivický

Pudlák

2017

Mathematical Logic Qtrly

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ABSTRACT. Linear arithmetics are extensions of Presburger arithmetic (Pr) by one or more unary functions, each intended as multiplication by a fixed element (scalar), and containing the full induction schemes for their respective languages.In this paper we construct a model M of the 2-linear arithmetic LA 2 (linear arithmetic with two scalars) in which an infinitely long initial segment of "Peano multiplication" on M is ∅-definable. This shows, in particular, that LA 2 is not model complete in contrast to theories LA 1 and LA 0 = Pr that are known to satisfy quantifier elimination up to disjunctions of primitive positive formulas.As an application, we show that M, as a discretely ordered module over the discretely ordered ring generated by the two scalars, is not NIP, answering negatively a question of Chernikov and Hils.

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“…The same properties—bounded pp‐elimination , (consequently) model completeness and decidability (cf. and independently )—have been be shown also for