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Analytic determinacy and 0#

  • Leo Harrington (a1)

Martin [12] has shown that the determinacy of analytic games is a consequence of the existence of sharps. Our main result is the converse of this:

Theorem. If analytic games are determined, then x2 exists for all reals x.

This theorem answers question 80 of Friedman [5]. We actually obtain a somewhat sharper result; see Theorem 4.1. Martin had previously deduced the existence of sharps from 3 − Π11-determinacy (where α − Π11 is the αth level of the difference hierarchy based on − Π11 see [1]). Martin has also shown that the existence of sharps implies < ω2 − Π11-determinacy.

Our method also produces the following:

Theorem. If all analytic, non-Borel sets of reals are Borel isomorphic, then x* exists for all reals x.

The converse to this theorem had been previously proven by Steel [7], [18].

We owe a debt of gratitude to Ramez Sami and John Steel, some of whose ideas form basic components in the proofs of our results.

For the various notation, definitions and theorems which we will assume throughout this paper, the reader should consult [3, §§5, 17], [8], [13] and [14, Chapter 16].

Throughout this paper we will concern ourselves only with methods for obtaining 0# (rather than x# for all reals x). By relativizing our arguments to each real x, one can produce x2.

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[1]Addison J.W., The theory of hierarchies, Proceedings of the 1960 International Congress of Logic, Methodology and Philosophy of Science Stanford University Press, Stanford, 1960, pp. 2637.
[2]Baumgartner J.E., Harrington L.A. and Kleinberg E.M., Adding a closed unbounded set, this Journal, vol. 41 (1976), pp. 481482.
[3]Devlin K.J., Aspects of constructibility, Lecture Notes in Mathematics, no. 354, Springer-Verlag, Berlin and New York, 1973.
[4]Friedman H., Determinacy in the low projective hierarchy, Fundamenta Mathematicae, vol. 72(1971), pp. 7984.
[5]Friedman H., One hundred and two problems in mathematical logic, this Journal, vol. 40 (1975), pp. 113129.
[6]Harrington L.A., Σ11sets and hyperdegrees (in preparation).
[7]Harrington L.A. and Steel J., Analytic sets and Borel isomorphism, Notices of the American Mathematical Society, vol. 23 (1976), p. A447.
[8]Jech T.J., Lectures in set theory, Lecture Notes in Mathematics, no. 217, Springer-Verlag, Berlin and New York, 1971.
[9]Kechris A.S., and Moschovakis Y.N., Recursion in higher types, Handbook for logic, North-Holland, Amsterdam, 1976.
[10]Keisler H.J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.
[11]Martin D.A., The axiom of determinacy and reduction principles in the analytical hierarchy. Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687689.
[12]Martin D.A., Measurable cardinals and analytic games, Fundamenta Mathematicae, vol. 66 (1970), pp. 287291.
[13]Martin D.A., Descriptive set theory: Projective sets, Handbook for logic, North-Holland, Amsterdam, 1976.
[14]Rogers H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[15]Kunen K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 3 (1970), pp. 179227.
[16]Simpson S., Minimal covers and hyperdegrees, Transactions of the American Mathematical Society, vol. 209 (1975), pp. 4564.
[17]Steel J., Ph.D. Thesis, University of California, Berkeley, 1976.
[18]Steel J., Analytic sets and Borel isomorphisms (to appear).
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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