Skip to main content
×
Home
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 35
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Finkel, Olivier 2016. Infinite games specified by 2-tape automata. Annals of Pure and Applied Logic,


    Hölzl, Rupert Stephan, Frank and Yu, Liang 2016. On Martin’s pointed tree theorem. Computability, Vol. 5, Issue. 2, p. 147.


    Cheng, Yong 2015. The strong reflecting property and Harrington's Principle. Mathematical Logic Quarterly, Vol. 61, Issue. 4-5, p. 329.


    Cheng, Yong 2015. Forcing a set model of Z3 + Harrington's Principle. Mathematical Logic Quarterly, Vol. 61, Issue. 4-5, p. 274.


    MONTALBÁN, ANTONIO 2015. ANALYTIC EQUIVALENCE RELATIONS SATISFYING HYPERARITHMETIC-IS-RECURSIVE. Forum of Mathematics, Sigma, Vol. 3,


    Rittberg, Colin J. 2015. How Woodin changed his mind: new thoughts on the Continuum Hypothesis. Archive for History of Exact Sciences, Vol. 69, Issue. 2, p. 125.


    Schlicht, Philipp 2014. Thin equivalence relations and inner models. Annals of Pure and Applied Logic, Vol. 165, Issue. 10, p. 1577.


    Arrigoni, Tatiana and Friedman, Sy-David 2012. Foundational implications of the Inner Model Hypothesis. Annals of Pure and Applied Logic, Vol. 163, Issue. 10, p. 1360.


    Larson, Paul B. 2012. Sets and Extensions in the Twentieth Century.


    Yu, Liang 2011. A New Proof of Friedman's Conjecture. Bulletin of Symbolic Logic, Vol. 17, Issue. 03, p. 455.


    Andretta, Alessandro 2006. More on Wadge determinacy. Annals of Pure and Applied Logic, Vol. 144, Issue. 1-3, p. 2.


    Andretta, Alessandro 2003. Equivalence between Wadge and Lipschitz determinacy. Annals of Pure and Applied Logic, Vol. 123, Issue. 1-3, p. 163.


    NEEMAN, ITAY 2002. OPTIMAL PROOFS OF DETERMINACY II. Journal of Mathematical Logic, Vol. 02, Issue. 02, p. 227.


    Becker, Howard 2001. Topics in invariant descriptive set theory. Annals of Pure and Applied Logic, Vol. 111, Issue. 3, p. 145.


    Becker, Howard and Dougherty, Randall 1999. On Disjoint Borel Uniformizations. Advances in Mathematics, Vol. 146, Issue. 2, p. 167.


    Hjorth, Greg 1999. Orbit cardinals: On the effective cardinalities arising as quotient spaces of the formX/G whereG acts on a Polish spaceX. Israel Journal of Mathematics, Vol. 111, Issue. 1, p. 221.


    Hjorth, Greg 1996. Π12 Wadge degrees. Annals of Pure and Applied Logic, Vol. 77, Issue. 1, p. 53.


    Hjorth, Greg 1996. Two Applications of Inner Model Theory to the Study of Sets. Bulletin of Symbolic Logic, Vol. 2, Issue. 01, p. 94.


    Welch, P.D. 1996. Determinacy in the difference hierarchy of co-analytic sets. Annals of Pure and Applied Logic, Vol. 80, Issue. 1, p. 69.


    Neeman, Itay 1995. Optimal Proofs of Determinacy. Bulletin of Symbolic Logic, Vol. 1, Issue. 03, p. 327.


    ×

Analytic determinacy and 0#

  • Leo Harrington (a1)
  • DOI: http://dx.doi.org/10.2307/2273508
  • Published online: 01 March 2014
Abstract

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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[11]D.A. Martin , The axiom of determinacy and reduction principles in the analytical hierarchy. Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687689.

[16]S. Simpson , Minimal covers and hyperdegrees, Transactions of the American Mathematical Society, vol. 209 (1975), pp. 4564.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×