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THE RAMIFIED ANALYTICAL HIERARCHY USING EXTENDED LOGICS

Published online by Cambridge University Press:  25 October 2018

PHILIP D. WELCH
PHILIP D. WELCH*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF BRISTOL BRISTOL, BS8 1TW, UKE-mail: p.welch@bristol.ac.uk
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Abstract

The use of Extended Logics to replace ordinary second order definability in Kleene’s Ramified Analytical Hierarchy is investigated. This mirrors a similar investigation of Kennedy, Magidor and Väänänen [11] where Gödel’s universe L of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain minimal correct models of analysis. A wide spectrum of models can be so generated from abstract definability notions: one may take an abstract Spector Class and extract an extended logic for it. The resultant structure is then a minimal model of the given kind of definability.

Keywords

03E6003E1503E47definabilityanalytical hierarchydeterminacy
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 24 , Issue 3 , September 2018 , pp. 306 - 318
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Aczel, P., Quantifiers, games and inductive definitions, Proceedings of the Third Scandinavian Logic Symposium (University of Uppsala, Uppsala, 1973) (Kanger, S., editor), Studies in Logic and the Foundations of Mathematics, vol. 82, North-Holland, Amsterdam, 1975, pp. 1–14.CrossRefGoogle Scholar
Boolos, G. and Putnam, H., Degrees of unsolvability of constructible sets of integers. The Journal for Symbolic Logic, vol. 33 (1968), pp. 497–513.CrossRefGoogle Scholar
Boyd, R., Hensel, G., and Putnam, H., A recursion-theoretic characterization of the ramified analytical hierarchy. Transactions of the American Mathematical Society, vol. 141 (1966), pp. 37–62.CrossRefGoogle Scholar
Cohen, P., A minimal model for set theory. Bulletin of the American Mathematical Society, vol. 69 (1963), pp. 537–540.CrossRefGoogle Scholar
Enderton, H. and Friedman, H., Approximating the standard model of analysis. Fundamenta Mathematicae, vol. 72 (1971), no. 2, pp. 175–188.CrossRefGoogle Scholar
Friedman, H., Higher set theory and mathematical practice. Annals of Mathematical Logic, vol. 2 (1970), no. 3, pp. 325–327.CrossRefGoogle Scholar
Kechris, A., Martin, D. A., and Solovay, R., Introduction to Q-theory, Cabal Seminar 81–85 (Kechris, A., Martin, D. A., and Moschovakis, Y., editors), Lecture Notes in Mathematics, vol. 1333, Springer Verlag, Berlin, 1988, pp. 199–281.CrossRefGoogle Scholar
Kechris, A. S., On Spector classes, Cabal Seminar 76–77 (Kechris, A. S. and Moschovakis, Y. N., editors), Lecture Notes in Mathematics Series, vol. 689, Springer, Berlin, 1978, pp. 245–278.CrossRefGoogle Scholar
Kennedy, J., On formalism freeness: Implementing Gödel’s 1946 princeton bicentennial lecture, this Bulletin, vol. 19 (2013), no. 3, pp. 351–393.Google Scholar
Kennedy, J., Turing, Gödel and the “Bright Abyss”. Boston Studies in Philosophy and History of Science, vol. 324 (2017), pp. 63–91.CrossRefGoogle Scholar
Kennedy, J., Magidor, M., and Väänänen, J., Inner models from extended logics. Isaac Newton Preprint Series, vol. NI 16006 (2016), pp. 1–75.Google Scholar
Kleene, S. C., Recursive quantifiers and functionals of finite type I. Transactions of the American Mathematical Society, vol. 91 (1959), pp. 1–52.Google Scholar
Moschovakis, Y. N., Uniformization in a playful universe. Bulletin of the American Mathematical Society, vol. 77 (1971), pp. 731–736.CrossRefGoogle Scholar
Moschovakis, Y. N., Elementary Induction on Abstract Structures, Studies in Logic Series, vol. 77, North-Holland, Amsterdam, 1974.Google Scholar
Moschovakis, Y. N., Descriptive Set Theory, Studies in Logic Series, North-Holland, Amsterdam, 2009.CrossRefGoogle Scholar
Shilleto, J. R., Minimum models of analysis. Journal for Symbolic Logic, vol. 37 (1972), no. 1, pp. 48–54.CrossRefGoogle Scholar
Simpson, S., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer, Berlin, 1999.CrossRefGoogle Scholar
Steel, J. R., Projectively well-ordered inner models. Annals of Pure and Applied Logic, vol. 74 (1995), no. 1, pp. 77–104.CrossRefGoogle Scholar
Welch, P. D., Weak systems of determinacy and arithmetical quasi-inductive definitions. The Journal for Symbolic Logic, vol. 76 (2011), no. 2, pp. 418–436.CrossRefGoogle Scholar
Welch, P. D., Gδσ-games and generalized computation. Unpublished manuscript. Available at http://maths.bris.ac.uk/∼mapdw/Sigma0-3-IJ-Sep-2016.pdf.Google Scholar