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On a new notion of partial conservativity

Part of the Lecture Notes in Mathematics book series (LNM,volume 1104)

Abstract

Roughly, a bounded formula Φ(x) is (2C,c)-conservative if assuming Φ(2C) gives no new bounded information on c (c being a constant for a non-standard element in Peano arithmetic PA). Similarly for iterated powers of 2. This notion is analyzed, various existence theorems are proved and, as a corollary, we obtain a strengthening of Second Gödel's Incompleteness Theorem saying that for each non-standard model M of PA and each non-standard element a ε M there is a model K of PA coinciding with M up to a and such that in K there is a very short proof of constradiction (bounded by 2 to the 2 to the 2 to the c).

Keywords

  • Atomic Formula
  • Conservative Extension
  • Peano Arithmetic
  • Satisfaction Relation
  • Incompleteness Theorem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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© 1984 Springer-Verlag

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Hájek, P. (1984). On a new notion of partial conservativity. In: Börger, E., Oberschelp, W., Richter, M.M., Schinzel, B., Thomas, W. (eds) Computation and Proof Theory. Lecture Notes in Mathematics, vol 1104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099487

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  • DOI: https://doi.org/10.1007/BFb0099487

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13901-0

  • Online ISBN: 978-3-540-39119-7

  • eBook Packages: Springer Book Archive