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Fragments of Heyting arithmetic

Published online by Cambridge University Press:  12 March 2014

Wolfgang Burr
Wolfgang Burr*
Affiliation:
Institut für Mathematische Logik und Grundlagenforschung, der Westfälischen Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany E-mail: [email protected]
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Abstract

We define classes Φn of formulae of first-order arithmetic with the following properties:

(i) Every φ ϵ Φn is classically equivalent to a Πn-formula (n ≠ 1, Φ1 := Σ1).

(ii)

(iii) IΠn and iΦn (i.e., Heyting arithmetic with induction schema restricted to Φn-formulae) prove the same Π2-formulae.

We further generalize a result by Visser and Wehmeier. namely that prenex induction within intuitionistic arithmetic is rather weak: After closing Φn both under existential and universal quantification (we call these classes Θn) the corresponding theories iΘn still prove the same Π2-formulae. In a second part we consider iΔ0 plus collection-principles. We show that both the provably recursive functions and the provably total functions of are polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 3 , September 2000 , pp. 1223 - 1240
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Avigad, Jeremy, Interpreting classical theories in constructive ones, preprint, 1998.Google Scholar
[2]Avigad, Jeremy and Sommer, Rick, A model-theoretic approach to ordinal analysis, The Bulletin of Symbolic Logic, vol. 3 (1997), pp. 17–52.CrossRefGoogle Scholar
[3]Beklemishev, Lev D., A proof-theoretic analysis of collection, Archive for Mathematical Logic, vol. 37 (1998), pp. 275–296.CrossRefGoogle Scholar
[4]Buss, Samuel, Bounded arithmetic, Bibliopolis, Naples, 1986, revision of 1985 Princeton University Ph. D. thesis.Google Scholar
[5]Coquand, Thierry and Hofmann, Martin, A new method for establishing conservativity of classical systems over their intuitionistic version, Lambda calculus and logic, Mathematical Structures in Computer Science, vol. 9 (1999), no. 4, pp. 323–333.CrossRefGoogle Scholar
[6]Diller, Justus and Nahm, Werner, Eine Variante zur Dialectica-Interpretation der Heyting-Arithmetik endlicher Typen, Archiv für mathematische Logik und Grundlagenforschung, vol. 16 (1974), pp. 49–66.CrossRefGoogle Scholar
[7]Diller, Justus and Schütte, Kurt, Simultane Rekursionen in der Theorie der Funktionale endlicher Typen, Archiv für mathematische Logik und Grundlagenforschung, vol. 14 (1970), pp. 69–74.Google Scholar
[8]Gödel, Kurt, Über eine bisher noch nicht benützte Erweiterung desfiniten Standpunktes, Dialectica, vol. 12 (1958), pp. 280–287.CrossRefGoogle Scholar
[9]Hájek, Petr and Pudlák, Pavel, Metamathematics of first-order arithmetic, Berlin, 1993.CrossRefGoogle Scholar
[10]Kaye, Richard, Models of Peano arithmetic, Oxford, 1991.CrossRefGoogle Scholar
[11]Kohlenbach, Ulrich, Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals, Archive for Mathematical Logic, vol. 36 (1996), pp. 31–71.CrossRefGoogle Scholar
[12]Leivant, Daniel, Implicational complexity in intuitionistic arithmetic, this Journal, vol. 46 (1981), pp. 240–248.Google Scholar
[13]Parsons, Charles, On n-quantifier induction, this Journal, vol. 37 (1972), pp. 466–482.Google Scholar
[14]Pohlers, Wolfram, Proof theory, Berlin, 1989.CrossRefGoogle Scholar
[15]Schütte, Kurt, Proof theory, Heidelberg, 1977.CrossRefGoogle Scholar
[16]Shoenfield, J. R., Mathematical logic, 2nd ed., Addison-Wesley, Reading, Massachusetts, 1973.Google Scholar
[17]Sieg, Wilfred, Fragments of arithmetic, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 33–71.CrossRefGoogle Scholar
[18]Troelstra, A. S. (editor), Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, 1973.CrossRefGoogle Scholar
[19]Troelstra, A. S., Aspects of constructive mathematics, Handbook of mathematical logic (Barwise, Jon, editor), North-Holland, 1977, pp. 973–1052.Google Scholar
[20]Troelstra, A. S. and van Dalen, D., Constructivism in mathematics, vol. I, North-Holland, Amsterdam, 1988.Google Scholar
[21]Wehmeier, Kai F., Semantical investigations in intuitionistic first-order arithmetic, Ph.D. thesis, Münster, 1996.Google Scholar
[22]Wehmeier, Kai F., Fragments of HA based on Σ1-induction, Archive for Mathematical Logic, vol. 37 (1997), pp. 445–460.CrossRefGoogle Scholar
[23]Weiermann, Andreas, How is it, that infinitary methods can be applied to finitary mathematics? Gödel's T; a case study, this Journal, vol. 63 (1998), pp. 1348–1370.Google Scholar