Abstract
In this paper, we prove results concerning the existence of proper end extensions of arbitrary models of fragments of Peano arithmetic (PA). In particular, we give alternative proofs that concern (a) a result of Clote (Fundam Math 127(2):163–170, 1986); (Fundam Math 158(3):301–302, 1998), on the end extendability of arbitrary models of \(\Sigma _n\)-induction, for \(n{\ge } 2\), and (b) the fact that every model of \(\Sigma _1\)-induction has a proper end extension satisfying \(\Delta _0\)-induction; although this fact was not explicitly stated before, it follows by earlier results of Enayat and Wong (Ann Pure Appl Log 168:1247–1252, 2017) and Wong (Proc Am Math Soc 144:4021–4024, 2016).
Similar content being viewed by others
References
Adamowicz, Z.: End-extending models of \(I\Delta _0 + exp + B\Sigma _1\). Fundam. Math. 136, 133–145 (1990)
Adamowicz, Z.: A contribution to the end-extension problem and the \(\Pi _1\) consrvativeness problem. Ann. Pure Appl. Log. 61, 3–48 (1993)
Clote, P.: A note on the MacDowell–Specker theorem. Fundam. Math. 127(2), 163–170 (1986)
Clote, P.: A note on the MacDowell–Specker theorem. Fundam. Math. 158(3), 301–302 (1998)
Clote, P., Krajiček, J.: Open problems, Oxford logic guides. In: Arithmetic, Proof Theory, and Computational Complexity (Prague, 1991), vol. 23, pp. 1–19. Oxford University Press, New York (1993)
Dimitracopoulos, C., Paschalis, V.: End extensions of models of weak arithmetic theories. Notre Dame J. Form. Log. 57, 181–193 (2016)
Dimitracopoulos, C., Paschalis, V.: End extensions of models of arithmetic theories, II. In: Proceedings of the 11th Panhellenic Logic Symposium (Delphi, 2017), pp. 226–231
Enayat, A., Wong, T.L.: Unifying the model theory of first-order and second-order arithmetic via \(WKL^*_0\). Ann. Pure Appl. Log. 168, 1247–1252 (2017)
Gaifman, H.: Models and types of Peano’s arithmetic. Ann. Math. Log. 9, 223–306 (1976)
Hájek, P.: Interpretability and fragments of arithmetic. In: Clote, P., Krajíček, J. (eds.) Arithmetic, Proof Theory and Computational Complexity, Oxford Logic Guides, vol. 23, pp. 185–196. Clarendon Press, Oxford (1993)
Hájek, P., Pudlák, P.: Metamathematics of First-Order Arithmetic. Springer, Berlin (1993)
Kaye, R.: Models of Peano Arithmetic. Oxford Logic Guides, vol. 15. Oxford University Press, New York (1991)
Kaye, R., Paris, J., Dimitracopoulos, C.: On parameter free induction schemas. J. Symb. Log. 53, 1082–1097 (1988)
MacDowell, R., Specker, E.: Modelle der Arithmetik. In: Infinitistic Methods. Proceedings of the Symposium on Foundations of Mathematics, pp. 257–263. Pergamon Press, Oxford (1961)
Paris, J.B.: Some Conservation Results for Fragments of Arithmetic, Model Theory and Arithmetic (Paris, 1979–1980). Lecture Notes in Mathematics, vol. 890, pp. 251–262. Springer, Berlin (1981)
Paris, J.B., Kirby, L.A.S.: \(\Sigma _{n}\)-collection schemas in arithmetic. In: Logic Colloquium ’77 (Proc. Conf., Wroclaw, 1977), pp. 199–209, North-Holland, Amsterdam, New York (1978)
Wilkie, A.J., Paris, J.B.: On the scheme of induction for bounded arithmetic formulas. Ann. Pure Appl. Log. 35(3), 261–302 (1987)
Wilkie, A., Paris, J.: On the Existence of End Extensions of Models of Bounded Induction, Logic, Methodology and Philosophy of Science, VIII (Moscow, 1987). Studies in Logic and the Foundations of Mathematics, vol. 126, pp. 143–161. Elsevier, Amsterdam (1989)
Wong, T.L.: Interpreting weak König’s lemma using the arithmetized completeness theorem. Proc. Am. Math. Soc. 144, 4021–4024 (2016)
Acknowledgements
The authors are grateful to Ali Enayat and Tin Lok Wong, for bringing [8, 10] and [19] to their attention, as well as for helpful comments on earlier, shorter or wrong, versions of the present paper. They are also grateful to the unknown referee(s), whose insightful corrections and remarks played a decisive role in producing the final version of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dimitracopoulos, C., Paschalis, V. End extensions of models of fragments of PA. Arch. Math. Logic 59, 817–833 (2020). https://doi.org/10.1007/s00153-019-00708-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-019-00708-4