Archive for Mathematical Logic

, Volume 55, Issue 7–8, pp 899–924 | Cite as

Fragments of Kripke–Platek set theory and the metamathematics of \(\alpha \)-recursion theory

Open Access
Article

Abstract

The foundation scheme in set theory asserts that every nonempty class has an \(\in \)-minimal element. In this paper, we investigate the logical strength of the foundation principle in basic set theory and \(\alpha \)-recursion theory. We take KP set theory without foundation (called KP\(^-\)) as the base theory. We show that KP\(^-\) + \(\Pi _1\)-Foundation + \(V=L\) is enough to carry out finite injury arguments in \(\alpha \)-recursion theory, proving both the Friedberg-Muchnik theorem and the Sacks splitting theorem in this theory. In addition, we compare the strengths of some fragments of KP.

Keywords

Metamathematics Foundation Kripke–Platek set theory \(\alpha \)-recursion theory 

Mathematics Subject Classification

03D65 

References

  1. 1.
    Ackermann, W.: Die widerspruchsfreiheit der allgemeinen mengenlehre. Math. Ann. 114(1), 305–315 (1937)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barwise, J.: Admissible Sets and Structures. Springer, Berlin (1975)CrossRefMATHGoogle Scholar
  3. 3.
    Cantini, A.: Nonextensional theories of predicative classes over PA. Rend. Sem. Mat. Univ. Politec. Torino 40(3), 47–79 (1984)MathSciNetMATHGoogle Scholar
  4. 4.
    Cantini, A.: A note on the theory of admissible sets with \(\in \)-induction restricted to formulas with one quantifier and related systems. Boll. Un. Mat. Ital. B 2(3), 721–737 (1983)MathSciNetMATHGoogle Scholar
  5. 5.
    Chong, C.T., Mourad, K.J.: The degree of a \(\Sigma _n\) cut. Ann. Pure Appl. Log. 48, 227–235 (1990)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Friedman, H.: Countable models of set theories. In: Mathias, A.R.D., Rogers, H. (eds.) Cambridge Summer School in Mathematical Logic. Lecture Notes in Mathematics, vol. 337, pp. 539–573. Springer, Berlin, Heidelberg (1973)Google Scholar
  7. 7.
    Friedman, S.D.: \(\beta \)-recursion theory. Trans. Am. Math. Soc. 255, 173–200 (1979)MathSciNetMATHGoogle Scholar
  8. 8.
    Groszek, M.J., Mytilinaios, M.E., Slaman, T.A.: The Sacks density theorem and \(\Sigma _2\)-bounding. J. Symb. Log. 61(2), 450–467 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jech, T.: Set Theory: The Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics, 3rd edn. Springer, Berlin (2003)Google Scholar
  10. 10.
    Kaye, R., Wong, T.L.: On interpretations of arithmetic and set theory. Notre Dame J. Form. Log. 48(4), 497–510 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kleene, S.C.: On the form of predicates in the theory of constructive ordinals (second paper). Am. J. Math. 77, 405–428 (1955)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lerman, M., Sacks, G.E.: Some minimal pairs of \(\alpha \)-recursively enumerable degrees. Ann. Math. Log. 4, 415–442 (1972)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lubarsky, R.S.: An introduction to \(\gamma \)-recursion theory (or what to do in \({\rm KP}\,-\,\) foundation). J. Symb. Log. 55(1), 194–206 (1990)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Maass, W.: On minimal pairs and minimal degrees in higher recursion theory. Arch. Math. Log. Grundlag. 18(3–4), 169–186 (1976)MathSciNetMATHGoogle Scholar
  15. 15.
    Mytilinaios, M.: Finite injury and \(\Sigma _1\)-induction. J. Symb. Log. 54(1), 38–49 (1989)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mytilinaios, M.E., Slaman, T.A.: \(\Sigma _2\)-collection and the infinite injury priority method. J. Symb. Log. 53(1), 212–221 (1988)MathSciNetMATHGoogle Scholar
  17. 17.
    Paris, J.B., Kirby, L.A.S.: \(\Sigma _{n}\)-collection schemas in arithmetic. In: Logic Colloquium ’77 (Proceedings of Conference on Wrocław 1977), vol. 96 of Studies in Logic and the Foundations of Mathematics, pp. 199–209. North-Holland (1978)Google Scholar
  18. 18.
    Rathjen, M.: Fragments of Kripke–Platek set theory with infinity. In: Aczel, P., Simmons, H., Wainer, S.S. (eds.) Proof Theory (Leeds, 1990), pp. 251–274. Cambridge University Press, Cambridge (1993)Google Scholar
  19. 19.
    Rathjen, M.: A proof-theoretic characterization of the primitive recursive set functions. J. Symbol. Log. 57(3), 954–969 (1992)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Rathjen, M.: The natural numbers in constructive set theory. MLQ Math. Log. Q. 54(1), 83–97 (2008)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ressayre, J.P.: Modèles non standard et sous-systèmes remarquables de \(\rm {ZF}\). In: Modèles Non standard en Arithmétique et théorie des ensembles, vol. 22 of Publications Mathématiques de l’Université Paris VII, pp. 47–147. Université de Paris VII, U.E.R. de Mathématiques, Paris (1987)Google Scholar
  22. 22.
    Sacks, G.E., Simpson, S.G.: The \(\alpha \)-finite injury method. Ann. Math. Log. 4, 343–367 (1972)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Shore, R.A.: Splitting an \(\alpha \)-recursively enumerable set. Trans. Am. Math. Soc. 204, 65–77 (1975)MathSciNetMATHGoogle Scholar
  24. 24.
    Shore, R.A.: The recursively enumerable \(\alpha \)-degrees are dense. Ann. Math. Log. 9(1–2), 123–155 (1976)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Shore, R.A.: Some more minimal pairs of \(\alpha \)-recursively enumerable degrees. Z. Math. Log. Grundlag. Math. 24(5), 409–418 (1978)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Slaman, T.A.P.: \(\Sigma _n\)-bounding and \(\Delta _n\)-induction. Proc. Am. Math. Soc. 132(8), 2449–2456 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Kurt Gödel Research Center for Mathematical LogicUniversity of ViennaViennaAustria

Personalised recommendations