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Induction and inductive definitions in fragments of second order arithmetic

  • Klaus Aehlig (a1)

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.

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[1] 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. 739782.

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[4] 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. 2228.

[5] T. Arai , A slow growing analogue to Buchholz' proof, Annals of Pure and Applied Logic, vol. 54 (1991), pp. 101120.

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[12] D. Leivant , Finitely stratified polymorphism, Information and Computation, vol. 93 (1991), pp. 93113.

[15] G. Takeuti , Consistency proofs of subsystems of classical analysis, Annals of Mathematics, vol. 86 (1967), pp. 299348.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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