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| {{Orphan|date=September 2013}}
| | Friends call her Claude Gulledge. Her family life in Idaho. The preferred pastime for my children and me is playing crochet and now I'm trying to earn cash with it. Bookkeeping has been his working day occupation for a whilst.<br><br>Review my webpage [http://Discordia.Cwsurf.de/index.php?mod=users&action=view&id=61 extended auto warranty] |
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| In [[descriptive set theory]], the '''Martin measure''' is a [[filter (mathematics)|filter]] on the set of [[Turing degrees]] of sets of [[natural number]]s. Under the [[axiom of determinacy]] it can be shown to be an [[ultrafilter]].
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| == Definition ==
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| Let <math>D</math> be the set of Turing degrees of sets of natural numbers. Given some equivalence class <math>[X]\in D</math>, we may define the ''cone'' (or ''upward cone'') of <math>[X]</math> as the set of all Turing degrees <math>[Y]</math> such that <math>X\le_T Y</math>; that is, the set of Turing degrees which are "more complex" than <math>X</math> under Turing reduction.
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| We say that a set <math>A</math> of Turing degrees has measure 1 under the Martin measure exactly when <math>A</math> contains some cone. Since it is possible, for any <math>A</math>, to construct a game in which player I has a winning strategy exactly when <math>A</math> contains a cone and in which player II has a winning strategy exactly when the complement of <math>A</math> contains a cone, the [[axiom of determinacy]] implies that the measure-1 sets of Turing degrees form an ultrafilter.
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| == Consequences ==
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| It is easy to show that a countable intersection of cones is itself a cone; the Martin measure is therefore a [[Ultrafilter#Completeness|countably complete]] filter. This fact, combined with the fact that the Martin measure may be transferred to <math>\omega_1</math> by a simple mapping, tells us that <math>\omega_1</math> is measurable under the axiom of determinacy. This result shows part of the important connection between determinacy and [[large cardinal]]s.
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| == References ==
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| * {{cite book | author=Moschovakis, Yiannis N. | title=Descriptive Set Theory | publisher=North Holland | year=1980 |isbn=0-444-70199-0}}
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| [[Category:Descriptive set theory]]
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| [[Category:Determinacy]]
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| [[Category:Computability theory]]
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Latest revision as of 17:05, 25 December 2014
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