A wild model of linear arithmetic and discretely ordered modules

0484738 - MÚ 2018 RIV DE eng J - Journal Article
Glivický, Petr - Pudlák, Pavel
A wild model of linear arithmetic and discretely ordered modules.
Mathematical Logic Quarterly. Roč. 63, č. 6 (2017), s. 501-508. ISSN 0942-5616. E-ISSN 1521-3870
EU Projects: European Commission(XE) 339691 - FEALORA
Institutional support: RVO:67985840
Keywords : linear arithmetics
OECD category: Pure mathematics
Impact factor: 0.522, year: 2017

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 LA2 (linear arithmetic with two scalars) in which an infinitely long initial segment of Peano multiplication on M is phi-definable. This shows, in particular, that LA2 is not model complete in contrast to theories LA1 and LA0=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, does not have the NIP, answering negatively a question of Chernikov and Hils.
Permanent Link: http://hdl.handle.net/11104/0279883