Analytic determinacy and 0 #
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 x 2 exists for all reals x . This theorem answers question 80 of Friedman [5]. We actually obtain a somew...
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crcambridgeupr:10.2307/2273508 2024-09-15T18:33:33+00:00 Analytic determinacy and 0 # Harrington, Leo 1978 http://dx.doi.org/10.2307/2273508 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022481200049203 en eng Cambridge University Press (CUP) https://www.cambridge.org/core/terms Journal of Symbolic Logic volume 43, issue 4, page 685-693 ISSN 0022-4812 1943-5886 journal-article 1978 crcambridgeupr https://doi.org/10.2307/2273508 2024-07-03T04:04:22Z 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 x 2 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 − Π 1 1 -determinacy (where α − Π 1 1 is the αth level of the difference hierarchy based on − Π 1 1 see [1]). Martin has also shown that the existence of sharps implies < ω 2 − Π 1 1 -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 x 2 . Article in Journal/Newspaper sami Cambridge University Press Journal of Symbolic Logic 43 4 685 693 |
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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 x 2 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 − Π 1 1 -determinacy (where α − Π 1 1 is the αth level of the difference hierarchy based on − Π 1 1 see [1]). Martin has also shown that the existence of sharps implies < ω 2 − Π 1 1 -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 x 2 . |
format |
Article in Journal/Newspaper |
author |
Harrington, Leo |
spellingShingle |
Harrington, Leo Analytic determinacy and 0 # |
author_facet |
Harrington, Leo |
author_sort |
Harrington, Leo |
title |
Analytic determinacy and 0 # |
title_short |
Analytic determinacy and 0 # |
title_full |
Analytic determinacy and 0 # |
title_fullStr |
Analytic determinacy and 0 # |
title_full_unstemmed |
Analytic determinacy and 0 # |
title_sort |
analytic determinacy and 0 # |
publisher |
Cambridge University Press (CUP) |
publishDate |
1978 |
url |
http://dx.doi.org/10.2307/2273508 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022481200049203 |
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sami |
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sami |
op_source |
Journal of Symbolic Logic volume 43, issue 4, page 685-693 ISSN 0022-4812 1943-5886 |
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https://www.cambridge.org/core/terms |
op_doi |
https://doi.org/10.2307/2273508 |
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Journal of Symbolic Logic |
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43 |
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4 |
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685 |
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693 |
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1810475259689697280 |