Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-01T11:59:33.662Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on Σ1-maximal models

Published online by Cambridge University Press:  12 March 2014

A. Cordón-Franco ,
A. Fernández-Margarit  and
F.F. Lara-Martín
A. Cordón-Franco
Affiliation:
Dpto. Ciencias de la Computación e Inteligencia Artificial, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia, S/N, 41012 Sevilla, Spain, E-mail: acordon@us.es
A. Fernández-Margarit
Affiliation:
Dpto. Ciencias de la Computación e Inteligencia Artificial, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia, S/N, 41012 Sevilla, Spain, E-mail: afmargarit@us.es
F.F. Lara-Martín
Affiliation:
Dpto. Ciencias de la Computación e Inteligencia Artificial, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia, S/N, 41012 Sevilla, Spain, E-mail: fflara@us.es
Get access Rights & Permissions [Opens in a new window]

Abstract

Let T be a recursive theory in the language of first order Arithmetic. We prove that if T extends: (a) the scheme of parameter free Δ1-minimization (plus exp). or (b) the scheme of parameter free Π1-induction, then there are no Σ1-maximal models with respect to T. As a consequence, we obtain a new proof of an unpublished theorem of Jeff Paris stating that Σ1-maximal models with respect to IΔ0 + exp do not satisfy the scheme of Σ1-collection BΣ1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 1072 - 1078
Copyright
Copyright © Association for Symbolic Logic 2007

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

REFERENCES

[1]Adamowicz, Z., On maximal theories, this Journal, vol. 56 (1991), pp. 885–890.Google Scholar
[2]Adamowicz, Z. and Bigorajska, T., Existentially closed structures and Gödel's second incompleteness theorem, this Journal, vol. 66 (2001), pp. 349–356.Google Scholar
[3]Cordón-Franco, A., Fernández-Margarit, A., and Lara-Martín, F.F., Fragments of arithmetic and true sentences, Mathematical Logic Quarterly, vol. 51 (2005), pp. 313–328.CrossRefGoogle Scholar
[4]Hájek, P. and Pudlák, P., Metamathematics of first-order arithmetic, Perpectives in Mathematical Logic, Springer-Verlag, 1993.CrossRefGoogle Scholar
[5]Hirschfeld, J. and Wheeler, W., Forcing, arithmetic, division rings, Lecture Notes in Mathematics, vol. 454, Springer-Verlag, 1975.CrossRefGoogle Scholar
[6]Kaye, R., Models of Peano Arithmetic, Oxford Logic Guides, vol. 15, Oxford University Press, 1991.CrossRefGoogle Scholar
[7]Kaye, R., Paris, J., and Dimitracopoulos, C., On parameter free induction schemas, this Journal, vol. 53 (1988), pp. 1082–1097.Google Scholar
[8]Paris, J. and Kirby, L., Σn-collection schemas in arithmetic, Logic colloquium 77 (Macintyre, A., Pacholski, L., and Paris, J., editors), North-Holland, 1978, pp. 285–296.Google Scholar
[9]Wilkie, A. and Paris, J., On the existence of end extensions of models of bounded induction, Logic, methodology and philosophy of science VIII (Fenstad, J., Frolov, I., and Hilpinen, R., editors), North-Holland, 1989, pp. 143–161.Google Scholar