# Difference between revisions of "Simple set"

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− | A recursively-enumerable set of natural numbers (cf. [[ | + | A recursively-enumerable set of natural numbers (cf. [[Enumerable set]]) whose complement is an [[immune set]]. Simple sets are intermediate in the sense of [[Many-one reducibility|$m$-reducibility]] (cf. [[Recursive set theory]]) between [[solvable set]]s and [[creative set]]s. The latter are the largest among the enumerable sets in the sense of $m$-reducibility. Let $P$ be an arbitrary simple set, and let $K$ be an arbitrary creative set of natural numbers (e.g. the set of Gödel numbers of theorems of formal arithmetic); then there does not exist a [[general recursive function]] $f$ reducing $K$ to $P$, i.e. such that |

$$ | $$ | ||

x \in K \Leftrightarrow f(x) \in P \ . | x \in K \Leftrightarrow f(x) \in P \ . | ||

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====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Uspenskii, "Leçons sur les fonctions calculables" , Hermann (1966) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967) pp. 164–165</TD></TR></table> | + | <table> |

+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Uspenskii, "Leçons sur les fonctions calculables" , Hermann (1966) (Translated from Russian)</TD></TR> | ||

+ | <TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russian)</TD></TR> | ||

+ | <TR><TD valign="top">[3]</TD> <TD valign="top"> H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967) pp. 164–165</TD></TR> | ||

+ | </table> | ||

+ | |||

+ | ====Comments==== | ||

+ | A set is ''hypersimple'' if it is recursively enumerable and its complement is a [[hyperimmune set]]. | ||

+ | |||

+ | ====References==== | ||

+ | <table> | ||

+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> Nies, André. Computability and randomness. Oxford Logic Guides 51. Oxford: Oxford University Press (2009). ISBN 978-0-19-923076-1. Zbl 1169.03034</TD></TR> | ||

+ | </table> | ||

{{TEX|done}} | {{TEX|done}} | ||

[[Category:Computability and recursion theory]] | [[Category:Computability and recursion theory]] |

## Latest revision as of 12:32, 17 January 2016

A recursively-enumerable set of natural numbers (cf. Enumerable set) whose complement is an immune set. Simple sets are intermediate in the sense of $m$-reducibility (cf. Recursive set theory) between solvable sets and creative sets. The latter are the largest among the enumerable sets in the sense of $m$-reducibility. Let $P$ be an arbitrary simple set, and let $K$ be an arbitrary creative set of natural numbers (e.g. the set of Gödel numbers of theorems of formal arithmetic); then there does not exist a general recursive function $f$ reducing $K$ to $P$, i.e. such that $$ x \in K \Leftrightarrow f(x) \in P \ . $$

Reducibility of $P$ to $K$ always takes place, but not one solvable set is reducible to $K$.

#### References

[1] | V.A. Uspenskii, "Leçons sur les fonctions calculables" , Hermann (1966) (Translated from Russian) |

[2] | A.I. Mal'tsev, "Algorithms and recursive functions" , Wolters-Noordhoff (1970) (Translated from Russian) |

[3] | H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967) pp. 164–165 |

#### Comments

A set is *hypersimple* if it is recursively enumerable and its complement is a hyperimmune set.

#### References

[a1] | Nies, André. Computability and randomness. Oxford Logic Guides 51. Oxford: Oxford University Press (2009). ISBN 978-0-19-923076-1. Zbl 1169.03034 |

**How to Cite This Entry:**

Simple set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Simple_set&oldid=34507