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| {{for|the Goodman–Myhill theorem in constructive set theory|Diaconescu's theorem}}
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| In [[recursion theory|computability theory]] the '''Myhill isomorphism theorem''', named after [[John Myhill]], provides a characterization for two [[numbering (computability theory)|numbering]]s to induce the same notion of computability on a set.
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| == Myhill isomorphism theorem ==
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| Sets ''A'' and ''B'' of [[natural number]]s are said to be '''[[computable isomorphism|recursively isomorphic]]''' if there is a [[total function|total]] [[computable function|computable]] [[bijection]] ''f'' from the set of natural numbers to itself such that ''f''(''A'') = ''B''.
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| A set ''A'' of natural numbers is said to be '''[[many-one reduction|one-one reducible]]''' to a set ''B'' if there is a total computable injection ''f'' on the natural numbers such that <math>f(A) \subseteq B</math> and <math>f(\mathbb{N} \setminus A) \subseteq \mathbb{N} \setminus B</math>.
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| '''Myhill's isomorphism theorem''' states that two sets ''A'' and ''B'' of natural numbers are recursively isomorphic if and only if ''A'' is one-reducible to ''B'' and ''B'' is one-reducible to ''A''. The theorem is proved by an effective version of the argument used for the [[Cantor–Bernstein–Schroeder theorem|Schroeder–Bernstein theorem]].
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| A corollary of Myhill's theorem is that two [[Numbering (computability theory)|total numberings]] are [[one equivalent numbering|one-equivalent]] if and only if they are [[computable isomorphism|computably isomorphic]].
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| == References ==
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| *{{citation
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| | last = Myhill | first = John | authorlink = John Myhill
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| | doi = 10.1002/malq.19550010205
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| | journal = Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
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| | mr = 0071379
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| | pages = 97–108
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| | title = Creative sets
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| | volume = 1
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| | year = 1955}}.
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| *{{citation
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| | last = Rogers | first = Hartley, Jr. | author-link = Hartley Rogers, Jr.
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| | edition = 2nd
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| | isbn = 0-262-68052-1
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| | location = Cambridge, MA
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| | mr = 886890
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| | publisher = MIT Press
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| | title = Theory of recursive functions and effective computability
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| | year = 1987}}.
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| [[Category:Computability theory]]
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Myrtle Benny is how I'm known as and I feel comfy when people use the full title. For many years I've been operating as a payroll clerk. South Dakota is where me and my spouse reside. Doing ceramics is what my family and I enjoy.
Stop by my web-site :: http://www.breda.nl/users/noeliadfebdftijfsdnt