Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-28T01:04:22.066Z Has data issue: false hasContentIssue false
  • English
  • Français

ELEMENTARY EQUIVALENCE IN POSITIVE LOGIC VIA PRIME PRODUCTS

Part of: Model theory

Published online by Cambridge University Press:  05 July 2023

TOMMASO MORASCHINI Open the ORCID record for TOMMASO MORASCHINI [Opens in a new window] ,
JOHANN J. WANNENBURG Open the ORCID record for JOHANN J. WANNENBURG [Opens in a new window]  and
KENTARO YAMAMOTO Open the ORCID record for KENTARO YAMAMOTO [Opens in a new window]
TOMMASO MORASCHINI
Affiliation:
DEPARTAMENT DE FILOSOFIA FACULTAT DE FILOSOFIA UNIVERSITAT DE BARCELONA (UB) CARRER MONTALEGRE 6, 08001 BARCELONA, SPAIN E-mail: tommaso.moraschini@ub.edu
JOHANN J. WANNENBURG
Affiliation:
ÚSTAV INFORMATIKY AKADEMIE VĚD ČESKÉ REPUBLIKY POD VODÁRENSKOU VĚŽÍ 2 182 07 PRAHA 8, THE CZECH REPUBLIC and SCHOOL OF MATHEMATICS UNIVERSITY OF THE WITWATERSRAND JOHANNESBURG, SOUTH AFRICA and DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS UNIVERSITY OF PRETORIA PRIVATE BAG X20, HATFIELD PRETORIA 0028, SOUTH AFRICA E-mail: jamie.wannenburg@up.ac.za
KENTARO YAMAMOTO*
Affiliation:
ÚSTAV INFORMATIKY AKADEMIE VĚD ČESKÉ REPUBLIKY POD VODÁRENSKOU VĚŽÍ 2 182 07 PRAHA 8, THE CZECH REPUBLIC
*
E-mail: yamamoto@cs.cas.cz
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce prime products as a generalization of ultraproducts for positive logic. Prime products are shown to satisfy a version of Łoś’s Theorem restricted to positive formulas, as well as the following variant of the Keisler Isomorphism Theorem: under the generalized continuum hypothesis, two models have the same positive theory if and only if they have isomorphic prime powers of ultrapowers.

Keywords

Keisler isomorphism theorempositive model theoryprime productpositively existentially closed modelh-inductive theory

MSC classification

Primary: 03C07: Basic properties of first-order languages and structures 03C10: Quantifier elimination, model completeness and related topics 03C20: Ultraproducts and related constructions
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 18
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ben Yaacov, I., Positive model theory and compact abstract theories . Journal of Mathematical Logic , vol. 3 (2003), pp. 85–118.CrossRefGoogle Scholar
Ben Yaacov, I., Simplicity in compact abstract theories . Journal of Mathematical Logic , vol. 2 (2003), pp. 163–191.CrossRefGoogle Scholar
Ben Yaacov, I. and Poizat, B., Fondements de la logique positive, this Journal, vol. 72 (2007), no. 4, pp. 1141–1162.Google Scholar
Blok, W. J. and Pigozzi, D., Algebraizable Logics , Memoirs of the American Mathematical Society, vol. 396, American Mathematical Society, Providence, 1989.Google Scholar
Chang, C. C. and Keisler, H. J., Model Theory , third ed., Studies in Logic, vol. 73, North-Holland, Amsterdam, 1990.Google Scholar
Eklof, P. C., Ultraproducts for algebraists , Handbook of Mathematical Logic , Studies in Logic and the Foundations of Mathematics, vol. 90, Elsevier, Amsterdam, 1977, pp. 105–137.CrossRefGoogle Scholar
Frayne, T., Morel, A. C., and Scott, D. S., Reduced direct products . Fundamenta Mathematicae , vol. 51 (1962/1963), pp. 195–228.CrossRefGoogle Scholar
Keisler, H. J., Ultraproducts and elementary classes . Indagationes Mathematicae , vol. 23 (1961), pp. 477–495.CrossRefGoogle Scholar
Keisler, H. J., Ultraproducts and saturated models . Indagationes Mathematicae , vol. 67 (1964), pp. 178–186.CrossRefGoogle Scholar
Kunen, K., Ultrafilters and independent sets . Transactions of the American Mathematical Society , vol. 172 (1972), pp. 299–306.CrossRefGoogle Scholar
Moraschini, T., Raftery, J. G., and Wannenburg, J. J., Singly generated quasivarieties and residuated structures . Mathematical Logic Quarterly , vol. 66 (2020), no. 2, pp. 150–172.CrossRefGoogle Scholar
Poizat, B., Univers positifs, this Journal, vol. 71 (2006), no. 3, pp. 969–976.Google Scholar
Poizat, B., Quelques effets pervers de la positivité . Annals of Pure and Applied Logic , vol. 161 (2010), no. 6, pp. 812–816.CrossRefGoogle Scholar
Poizat, B. and Yeshkeyev, A., Positive Jonsson theories . Logica Universalis , vol. 12 (2018), pp. 101–127.CrossRefGoogle Scholar
Shelah, S., Every two elementarily equivalent models have isomorphic ultrapowers . Israel Journal of Mathematics , vol. 10 (1971), pp. 224–233.CrossRefGoogle Scholar
Wroński, A., Overflow rules and a weakening of structural completeness , Rozwazania o Filozfii Prawdziwej. Jerezmu Perzanowskiemuw Darze , Uniwersytetu Jagiellońskiego, Kraków, 2009.Google Scholar