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Intuitionistic fixed point theories over set theories

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In this paper we show that the intuitionistic fixed point theory FiXi(T) over set theories T is a conservative extension of T if T can manipulate finite sequences and has the full foundation schema.

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References

  1. Arai T.: Consistency proof via pointwise induction. Arch. Math. Logic 37, 149–165 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arai T.: Some results on cut-elimination, provable well-orderings, induction and reflection. Ann. Pure Appl. Logic 95, 93–184 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arai T.: Non-elementary speed-ups in logic calculi. Math. Logic Q. 6, 629–640 (2008)

    Article  MathSciNet  Google Scholar 

  4. Arai T.: Intuitionistic fixed point theories over Heyting arithmetic. In: Feferman, S., Sieg, W. (eds.) Proofs Categories and Computations. Essays in honor of Grigori Mints, pp. 1–14. College Publications, King’s College London, London (2010)

    Google Scholar 

  5. Arai T.: Quick cut-elimination for strictly positive cuts. Ann. Pure Appl. Logic 162, 807–815 (2011)

    Article  MathSciNet  Google Scholar 

  6. Arai T.: Proof theory of weak compactness. J. Math. Logic 13, 1350003 (2013)

    Article  MathSciNet  Google Scholar 

  7. Arai, T.: Conservations of first-order reflections. J. Symb. Logic 79, 814–825 (2014)

  8. Arai, T.: Lifting proof theory to the countable ordinals : Zermelo–Fraenkel’s set theory. J. Symb. Logic 79, 25–354 (2014)

  9. Arai, T.: Proof theory of second order indescribability (in preparation)

  10. Avigad J.: On the relationship between ATR0 and \({\widehat{\rm ID}_{ < \omega}}\). J. Symb. Logic 61, 768–779 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Barwise J.: Admissible Sets and Structures. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  12. Beeson M.: Goodman’s theorem and beyond. Pac. J. Math. 84, 1–16 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Buchholz W.: \({{\it\Omega}_{\mu+1}}\)-rule. In: Buchholz, W., Feferman, S., Pohlers, W., Sieg, W. (eds.) Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Lecture Notes in Mathematics, vol. 897, pp. 188–233. Springer, Berlin (1981)

    Chapter  Google Scholar 

  14. Buchholz W.: Notation system for infinitary derivations. Arch. Math. Logic 30, 277–296 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Buchholz W.: An intuitionistic fixed point theory. Arch. Math. Logic 37, 21–27 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Feferman S.: Iterated inductive fixed-point theories: applications to Hancock’s conjecture. In: Metakides, G. (ed.) Patras Logic Symposion, pp. 171–196. North-Holland, Amsterdam (1982)

    Chapter  Google Scholar 

  17. Mints, G.E.: Quick cut-elimination for monotone cuts. In: Games, Logic, and Constructive Sets (Stanford, CA, 2000), CSLI Lecture Notes, vol. 161, pp. 75–83. CSLI Publications, Stanford (2003)

  18. Rüede C., Strahm T.: Intuitionistic fixed point theories for strictly positive operators. Math. Logic Q. 48, 195–202 (2002)

    Article  MATH  Google Scholar 

  19. Schindler R., Zeman M.: Fine structure. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 1, pp. 605–656. Springer, Berlin (2010)

    Chapter  Google Scholar 

  20. Takeuti, G.: Proof Theory, 2nd edn. North-Holland, Amsterdam (1987). Reprinted from Dover Publications (2013)

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Correspondence to Toshiyasu Arai.

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Arai, T. Intuitionistic fixed point theories over set theories. Arch. Math. Logic 54, 531–553 (2015). https://doi.org/10.1007/s00153-015-0426-y

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  • DOI: https://doi.org/10.1007/s00153-015-0426-y

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