Journal of Symbolic Logic

Induction and inductive definitions in fragments of second order arithmetic

Klaus Aehlig

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Abstract

A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n+1 nested second order quantifications and those are in such a way, that no second order variable occurs free in the scope of another second order quantifier. The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary formulae increases the strength by one inductive definition.

Article information

Source
J. Symbolic Logic Volume 70, Issue 4 (2005), 1087-1107.

Dates
First available in Project Euclid: 18 October 2005

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1129642116

Digital Object Identifier
doi:10.2178/jsl/1129642116

Mathematical Reviews number (MathSciNet)
MR2194238

Zentralblatt MATH identifier
1118.03054

Citation

Aehlig, Klaus. Induction and inductive definitions in fragments of second order arithmetic. Journal of Symbolic Logic 70 (2005), no. 4, 1087--1107. doi:10.2178/jsl/1129642116. http://projecteuclid.org/euclid.jsl/1129642116.


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References

  • P. Aczel An introduction to inductive definitions, Handbook of Mathematical Logic (J. Barwise, editor), Studies in Logic and the Foundations of Mathematics, vol. 90, chapter C.7, North-Holland Publishing Company,1977, pp. 739--782.
  • K. Aehlig, On fragments of analysis with strengths of finitely iterated inductive definitions, Ph.D. thesis, Ludwig-Maximilians-Universität München, July 2003.
  • K. Aehlig and J. Johannsen An elementary fragment of second-order lambda-calculus, ACM Transactions on Computational Logic, vol. 6 (2005), no. 2, pp. 468--480.
  • T. Altenkirch and T. Coquand A finitary subsystem of the polymorphic lambda-calculus, Proceedings of the 5th international conference on typed lambda caculi and applications (TLCA '01) (S. Abramsky, editor), Lecture Notes in Computer Science, vol. 2044, Springer Verlag, Berlin,2001, pp. 22--28.
  • T. Arai A slow growing analogue to Buchholz' proof, Annals of Pure and Applied Logic, vol. 54 (1991), pp. 101--120.
  • W. Buchholz The $\Omega_\mu+1$-rule, In Buchholz et al. [?], chapter IV, pp. 189--233.
  • W. Buchholz, S. Feferman, W. Pohlers, and W. Sieg Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, Lecture Notes in Mathematics, vol. 897, Springer-Verlag, Berlin,1981.
  • W. Buchholz and K. Schütte Proof theory of impredicative subsystems of analysis, Studies in Proof Theory, vol. 2, Bibliopolis, Naples,1988.
  • S. Feferman Formal theories for transfinite iterations of generalized inductive definitions, Intuitionism and proof theory (proceedings of the summer conference at Buffalo N.Y. 1968) (A. Kino, J. Myhill, and R. E. Vesley, editors), Studies in Logic and the Foundation of Mathematics, North-Holland Publishing Company, Amsterdam,1970, pp. 303--326.
  • A. Grzegorczyk Some classes of recursive functions, Rozprawy Matematyczne, vol. 4 (1953).
  • L. Kalmár Egyszer\Hu példa eldönthetetlen aritmetikai problémára, Matematikai és Fizikai Lapok, vol. 50 (1943), pp. 1--23.
  • D. Leivant Finitely stratified polymorphism, Information and Computation, vol. 93 (1991), pp. 93--113.
  • R. Matthes, Extensions of system $F$ by iteration and primitive recursion on monotone inductive types, Ph.D. thesis, Fakultät für Mathematik und Informatik der Ludwig-Maximilians-Universität München,May 1998.
  • W. W. Tait Intensional interpretations of functionals of finite type. I, Journal of Symbolic Logic, vol. 32 (1967), no. 2, pp. 198--212.
  • G. Takeuti Consistency proofs of subsystems of classical analysis, Annals of Mathematics, vol. 86 (1967), pp. 299--348.