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On Some Semi-constructive Theories Related to Kripke–Platek Set Theory

  • Fernando Ferreira
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 13)

Abstract

We consider some very robust semi-constructive theories related to Kripke–Platek set theory, with and without the powerset operation. These theories include the law of excluded middle for bounded formulas, a form of Markov’s principle, the unrestricted collection scheme and, also, the classical contrapositive of the bounded collection scheme. We analyse these theories using forms of a functional interpretation which work in tandem with the constructible hierarchy (or the cumulative hierarchy, if the powerset operation is present). The main feature of these functional interpretations is to treat bounded quantifications as “computationally empty.” Our analysis is extended to a second-order setting enjoying some forms of class comprehension, including strict-\(\Pi ^1_1\) reflection. The key idea of the extended analysis is to treat second-order (class) quantifiers as bounded quantifiers and strict-\(\Pi ^1_1\) reflection as a form of collection. We will be able to extract some effective bounds from proofs in these systems in terms of the constructive tree ordinals up to the Bachmann–Howard ordinal.

Keywords

Intuitionistic Kripke–Platek set theory Functional interpretations \(\Sigma \)-ordinal Strict-\(\Pi ^1_1\) reflection Power Kripke–Platek set theory 

MSC (2010)

03F10 03F35 03F65 

References

  1. 1.
    Avigad, J., Feferman, S.: Gödel’s functional (“Dialectica”) interpretation. In: Buss, S.R. (ed.) Handbook of Proof Theory. Studies in Logic and the Foundations of Mathematics, vol. 137, pp. 337–405. North Holland, Amsterdam (1998)CrossRefGoogle Scholar
  2. 2.
    Avigad, J., Towsner, H.: Functional interpretation and inductive definitions. J. Symb. Log. 74(4), 1100–1120 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barwise, J.: Implicit definability and compactness in infinitary languages. In: Barwise, J. (ed.) The Syntax and Semantics of Infinitary Logic. Lecture Notes in Mathematics, vol. 72, pp. 1–35. Springer, Berlin (1968)CrossRefGoogle Scholar
  4. 4.
    Barwise, J.: Application of strict \(\Pi ^1_1\) predicates to infinitary logic. J. Symb. Log. 34, 409–423 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barwise, J.: Admissible Set Theory and Structures: An Approach to Definability Theory. Perspectives in Mathematical Logic. Springer, Berlin (1975)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bishop, E.: Schizophrenia in contemporary mathematics. In: Bishop, E. (ed.) Reflections on Him and His Research. Contemporary Mathematics, pp. 1–32. American Mathematical Society (1985). First published in 1973Google Scholar
  7. 7.
    Bridges, D.S., Richman, F.: Varieties of Constructive Mathematics, vol. 97. London Mathematical Society Lecture Notes Series. Cambridge University Press (1987)Google Scholar
  8. 8.
    Feferman, S.: On the strength of some semi-constructive theories. In: Feferman, S., Sieg, W. (eds.) Proof Categories and Computation: Essays in honor of Grigori Mints, pp. 109–129. College Publications, London (2010)Google Scholar
  9. 9.
    Feferman, S.: Logic, mathematics, and conceptual structuralism. In: Rush, P. (ed.) The Metaphysics of Logic, pp. 72–92. Cambridge University Press (2014)Google Scholar
  10. 10.
    Ferreira, F.: A new computation of the \(\Sigma \)-ordinal of KP\(\omega \). J. Symb. Log. 79, 306–324 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ferreira, F., Oliva, P.: Bounded functional interpretation. Ann. Pure Appl. Log. 135, 73–112 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Friedman, H.: Countable models of set theories. In: Mathias, A.R.D., Rogers Jr., H. (eds.) Cambridge Summer School in Mathematical Logic. Lecture Notes in Mathematics, vol. 337, pp. 539–573. Springer, Berlin (1973)CrossRefGoogle Scholar
  13. 13.
    Hindley, J.R., Seldin, J.P.: Introduction to Combinators and \(\lambda \)-Calculus, vol. 1. London Mathematical Society Student Texts. Cambridge University Press (1986)Google Scholar
  14. 14.
    Howard, W.A.: A system of abstract constructive ordinals. J. Symb. Log. 37(2), 355–374 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. Springer, Berlin (2008)zbMATHGoogle Scholar
  16. 16.
    Pozsgay, L.: Liberal intuitionism as a basis for set theory. In: Scott, D. (ed.) Axiomatic Set Theory (Part 1), vol. XIII. Proceedings of Symposia in Pure Mathematics, pp. 321–330. American Mathematical Society (1971)Google Scholar
  17. 17.
    Rathjen, M.: Relativized ordinal analysis: the case of Power Kripke-Platek set theory. Ann. Pure Appl. Log. 165, 316–339 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sacks, G.: Higher Recursion Theory. Perspectives in Mathematical Logic. Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar
  19. 19.
    Salipante, V.: On the consistency strenght of the strict \(\Pi ^1_1\) reflection principle. Ph.D. thesis, Universität Bern (2005)Google Scholar
  20. 20.
    Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Faculdade de Ciências, Departamento de MatemáticaUniversidade de LisboaLisboaPortugal

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