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{{wikify|date=July 2007}}
In [[recursion theory]], the mathematical theory of computability, a '''maximal set''' is a coinfinite [[recursively enumerable set|recursively enumerable subset]] ''A'' of the [[natural number|natural numbers]] such that for every further recursively enumerable subset ''B'' of the natural numbers, either ''B'' is [[cofinite]] or ''B'' is a finite variant of ''A'' or ''B'' is not a superset of ''A''. This gives an easy definition within the lattice of the recursively enumerable sets. Maximal sets have many interesting properties: they are [[simple set|simple]], [[hypersimple]], [[hyperhypersimple]]<!-- These terms should be defined and given context. --> and r-maximal; the latter property says that every recursive set ''R'' contains either only finitely many elements of the complement of ''A'' or almost all elements of the complement of ''A''. There are r-maximal sets that are not maximal; some of them do even not have maximal supersets. Myhill (1956)<!-- I DON'T SEE A PAPER BY MYHILL CITED HERE. --> asked whether maximal sets exists and Frieberg (1958) constructed one. Soare (1974)<!-- SAME COMMENT HERE. --> showed that the maximal sets form an orbit with respect to automorphism of the recursively enumerable sets under inclusion ([[modulo]] finite sets). On the one hand, every automorphism maps a maximal set ''A'' to another maximal set ''B''; on the other hand, for every two maximal sets ''A'', ''B'' there is an automorphism of the recursively enumerable sets such that ''A'' is mapped to ''B''.
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