Skip to main content Accessibility help
×
×
Home

Truth definitions without exponentiation and the Σ1 collection scheme

  • Zofia Adamowicz (a1), Leszek Aleksander Kołodziejczyk (a2) and J. Paris (a3)

Abstract

We prove that:

  1. • if there is a model of IΔ0 + ¬exp with cofinal Σ1-definable elements and a Σ1 truth definition for Σ1 sentences, then IΔ0 + ¬exp + ¬BΣ1 is consistent,
  2. • there is a model of IΔ0 + Ω1 + ¬exp with cofinal Σ1-definable elements, both a Σ2 and a Π2 truth definition for Σ1 sentences, and for each n ≥ 2, a Σ n truth definition for Σ n sentences.

The latter result is obtained by constructing a model with a recursive truth-preserving translation of Σ1 sentences into boolean combinations of sentences.

We also present an old but previously unpublished proof of the consistency of IΔ0 + ¬exp + ¬BΣ1 under the assumption that the size parameter in Lessan's Δ0 universal formula is optimal. We then discuss a possible reason why proving the consistency of IΔ0 + ¬exp + ¬BΣ1 unconditionally has turned out to be so difficult.

Copyright

References

Hide All
[AKZ03] Adamowicz, Z., Kołodziejczyk, L. A., and Zbierski, P., An application of a reflection principle, Fundamenta Mathematicae, vol. 180 (2003), pp. 139159.
[Bus95] Buss, S. R., Relating the bounded arithmetic and polynomial time hierarchies, Annals of Pure and Applied Logic, vol. 75 (1995), pp. 6777.
[HP93] Hájek, P. and Pudlák, P., Metamathematics of first-order arithmetic, Springer-Verlag, 1993.
[KP78] Kirby, L. A. S. and Paris, J. B., Σn collection schemas in arithmetic. Logic colloquium 77 (Macintyre, A.. Pacholski, L., and Paris, J., editors), Studies in Logic and the Foundations of Mathematics, vol. 96, North-Holland, 1978, pp. 199209.
[Les78] Lessan, H., Models of arithmetic, Ph.D. thesis, University of Manchester, 1978.
[WP89] Wilkie, A. J. and Paris, J. B., On the existence of end-extensions of models of bounded induction, Logic, methodology, and philosophy of science VIII (Moscow 1987) (Fenstad, J. E., Frolov, I. T., and Hilpinen, R., editors), North-Holland, 1989, pp. 143161.
[Zam96] Zambella, D., Notes on polynomially bounded arithmetic, this Journal, vol. 61 (1996), pp. 942966.
[Zam97] Zambella, D., End extensions of models of polynomially bounded arithmetic, Annals of Pure and Applied Logic, vol. 88 (1997), pp. 263277.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed