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End extensions of models of fragments of PA

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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).

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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.

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Correspondence to C. Dimitracopoulos.

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Dimitracopoulos, C., Paschalis, V. End extensions of models of fragments of PA. Arch. Math. Logic (2020). https://doi.org/10.1007/s00153-019-00708-4

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Keywords

  • Arithmetized completeness theorem
  • Fragments of Peano arithmetic
  • End extensions

Mathematics Subject Classification

  • Primary 03C62
  • Secondary 03F30
  • 03H15