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Iterated reflection principles and the ω-rule

Published online by Cambridge University Press:  12 March 2014

Ulf R. Schmerl
Ulf R. Schmerl*
Affiliation:
Der Ludwio-Maximilians-Universitat, D-8000 Munchen 2, Federal, Republic of Germany
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Extract

The ω-rule,

with the meaning “if the formula A(n) is provable for all n, then the formula ∀xA(x) is provable”, has a certain formal similarity with a uniform reflection principle saying “if A(n) is provable for all n, then ∀xA(x) is true”. There are indeed some hints in the literature that uniform reflection has sometimes been understood as a “formalized ω-rule” (cf. for example S. Feferman [1], G. Kreisel [3], G. H. Müller [7]). This similarity has even another aspect: replacing the induction rule or scheme in Peano arithmetic PA by the ω-rule leads to a complete and sound system PA∞, where each true arithmetical statement is provable. In [2] Feferman showed that an equivalent system can be obtained by erecting on PA a transfinite progression of formal systems PAα based on iterations of the uniform reflection principle according to the following scheme:

Then T = (∪dЄ, PAd, being Kleene's system of ordinal notations, is equivalent to PA∞. Of course, T cannot be an axiomatizable theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 4 , December 1982 , pp. 721 - 733
Copyright
Copyright © Association for Symbolic Logic 1982

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References

REFERENCES

[1]Feferman, S., Systems of predicative analysis, this Journal, vol. 29 (1964), pp. 1–30.Google Scholar
[2]Feferman, S., Transfinite progressions of axiomatic theories, this Journal, vol. 27 (1962), pp. 259–316.Google Scholar
[3]Kreisel, G., Five notes on transfinite progressions, Technical Report No. 5, Applied Mathematics and Statistics Laboratory, Stanford University, California, 1962.Google Scholar
[4]Kreisel, G. and Lévy, A., Reflection principles and their use for establishing the complexity of axiomatic systems, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 97–142.CrossRefGoogle Scholar
[5]Lopez-Escobar, E. G. K., An extremely restricted ω-rule, Fundamenta Mathematicae, vol. 90 (1976), pp. 159–172.CrossRefGoogle Scholar
[6]Minc, G. E., Finite investigation of infinite derivations, Zapiski Naučnyh Seminarov Lenin-gradskogo Mathematičeskogo Instituto Akademii Nauk, vol. 49 (1975), pp. 67–122.Google Scholar
[7]Müller, G. H., Über die unendliche Induktion, Infinitistic methods, Pergamon, Oxford, 1961, pp. 75–95.Google Scholar
[8]Schmerl, U. R., Eine von Reflexionsformeln erzeugte Feinstruktur über Erweiterungen der primitiv rekursiven Arithmetik, Dissertation, Heidelberg, 1978.Google Scholar
[9]Schmerl, U. R., A fine structure generated by reflection formulas over primitive recursive arithmetic, Proceedings of the Logic Colloquium, Mons, 1978, North-Holland, Amsterdam, pp. 335–350.Google Scholar
[10]Schütte, K., Beweistheorie, Springer, Berlin, 1960.Google Scholar
[11]Schwichtenberg, H., Some applications of cut-elimination, Handbook of mathematical logic, North-Holland, Amsterdam, 1975, pp. 279–303.Google Scholar
[12]Sundholm, G., The omega-rule, a survey, Thesis, University of Oxford, 1978.Google Scholar