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The role of parameters in bar rule and bar induction

Published online by Cambridge University Press:  12 March 2014

Michael Rathjen
Michael Rathjen*
Affiliation:
Institut für Mathematische Logik und Grundlagenforschung, Westfälische Wilhelms-Universität Münster, W-4400 Münster, Germany
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Abstract

For several subsystems of second order arithmetic T we show that the proof-theoretic strength of T + (bar rule) can be characterized in terms of T + (bar induction)□, where the latter scheme arises from the scheme of bar induction by restricting it to well-orderings with no parameters. In addition, we demonstrate that , ACA0 + (bar rule) and ACA0 + (bar induction)□ prove the same -sentences.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 2 , June 1991 , pp. 715 - 730
Copyright
Copyright © Association for Symbolic Logic 1991

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References

REFERENCES

[1]Brown, D. and Simpson, S. G., Which set existence axioms are needed to prove the separable Hahn-Banach theorem, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 123–144.CrossRefGoogle Scholar
[2]Buchholz, W. and Schütte, K., Proof theory of impredicative subsystems of analysis, Bibliopolis, Napoli, 1988.Google Scholar
[3]Cantini, A., On the relation between choice and comprehension principles in second order arithmetic, this Journal, vol. 51 (1986), pp. 360–373.Google Scholar
[4]Feferman, S., Formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis, Intuitionism and proof theory (Kino, A.et al., editors), North-Holland, Amsterdam, 1970, pp. 303–325.Google Scholar
[5]Feferman, S., Theories of finite type related to mathematical practice, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 913–971.CrossRefGoogle Scholar
(6) Feferman, S., A more perspicuous formal system for predicativity, Konstruktionen versus Positionen, Vol. I, Walter de Gruyter, Berlin, 1978, pp. 68–93.CrossRefGoogle Scholar
[7]Feferman, S. and Jäger, G., Choice principles, the bar rule and autonomously iterated comprehension schemes in analysis, this Journal, vol. 48 (1983), pp. 63–70.Google Scholar
[8]Feferman, S. and Sieg, W., Iterated inductive definitions and subsystems of analysis, in Buchholz, W.et al., Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, Lecture Notes in Mathematics, vol. 897, Springer-Verlag, Berlin, 1981, pp. 16–77.Google Scholar
[9]Friedman, H., Systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematicians (Vancouver, 1974), Vol. 1, Canadian Mathematical Congress, Montréal, 1975, pp. 235–242.Google Scholar
[10]Friedman, H., Simpson, S. G. and Smith, R. L., Countable algebra and set existence axioms, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 141–181.CrossRefGoogle Scholar
[11]Girard, J.-Y., Proof theory and logical complexity, Bibliopolis, Napoli, 1987.Google Scholar
[12]Hinman, P. G., Recursion-theoretic hierarchies, Springer-Verlag, Berlin, 1978.CrossRefGoogle Scholar
[13]Kreisel, G., A survey of proof theory, this Journal, vol. 33, (1968), pp. 321–388.Google Scholar
[14]Jäger, G., Theories for admissible sets: A unifying approach to proof theory, Bibliopolis, Napoli, 1986.Google Scholar
[15]Kaye, R., Paris, J. and Dimitracopoulos, C., On parameter free induction schemas, this Journal, vol. 53 (1988), pp. 1082–1097.Google Scholar
[16]Rathjen, M., Untersuchungen zu Teilsystemen der Zahlentheorie zweiter Stufe und der Mengenlehre mit einer zwischen -CA und -CA + BI liegenden Beweisstärke, Report, Institut für Mathematische Logik und Grundlagenforschung, Westfalische Wilhelms-Universität Münster, Münster, 1989.Google Scholar
[17]Schütte, K., Proof theory, Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
[18]Schwichtenberg, H., Some applications of cut elimination, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 868–895.Google Scholar
[19]Takeuti, G., Proof theory, North-Holland, Amsterdam, 1975.Google Scholar