## There is no Fat Orbit (0)

Venue: | Ann. Pure Appl. Logic |

Citations: | 6 - 3 self |

### BibTeX

@ARTICLE{Downey_thereis,

author = {Rod Downey and Leo Harrington},

title = {There is no Fat Orbit},

journal = {Ann. Pure Appl. Logic},

year = {},

volume = {80},

pages = {277--289}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions.

### Citations

120 |
Recursively enumerable sets of positive integers and their decision problems
- Post
- 1944
(Show Context)
Citation Context ...ecursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions. 1 Introduction Ever since the classic paper of Post =-=[11]-=-, it has been recognized that there is an intrinsic interaction between the two fundamental structures of recursion theorysR, the uppersemilattice of recursively enumerable (computably enumerable) deg... |

54 |
Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
- Martin
- 1966
(Show Context)
Citation Context ...ets form a definable orbit and realize only sets of the complete m-degree (see Soare [14]), Soare's proof in [13] that the maximal sets form an orbit which realizes sets in each high degree by Martin =-=[10]-=-, and the Downey-Stob proof that hemimaximal sets, halfs of nontrivial splittings of maximal sets, form an orbit that realizes many degrees including all high degrees, all jump classes and arbitrarily... |

32 | Automorphisms of the Lattice of Recursively Enumerable Sets - Cholak - 1995 |

14 |
A non-inversion theorem for the jump operator
- Shore
- 1988
(Show Context)
Citation Context ...in 0 0 degree a ! 0 00 such that if b 0 = c 0 = a then there exist B 2 b and C 2 c with B automorphic with C. However this attractive conjecture fails. To see this, Cooper [1] and independently Shore =-=[12]-=- constructed a jump class of noncappable degrees. In particular there is a \Sigma 2 set A ! T ; 00 such that for all recursively enumerable sets C, if C 0 j T A, then C has promptly simple degree. The... |

6 |
Posts’s program and incomplete recursively enumerable sets
- Harrington, Soare
- 1991
(Show Context)
Citation Context ... 2 theoretical properties of R. The is, of course, a persistent intuition that the two are interlinked. A beautiful example of this phenomenon can be found in the recent paper of Harrington and Soare =-=[7]-=- where a elementary property P is given such that there exist nonrecursive recursively enumerable sets X such that P(X) holds yet P(X) cannot hold of a Turing complete recursively enumerable set. Natu... |

6 |
Variations on promptly simple sets
- Maass
- 1985
(Show Context)
Citation Context ...e degrees of both C and D are promptly simple. Therefore C j T b C and D j T c D for some recursively enumerable sets b C and c D which are both promptly simple and with semilow complements. By Maass =-=[9]-=-, there is an automorphism of the lattice of recursively enumerable sets taking b C to c D. It is not clear if a weakened form of the false conjecture might work for double jump classes. (One cannot g... |

3 |
A jump class of noncappable degrees
- Cooper
- 1989
(Show Context)
Citation Context ...ture (False)There is no REA in 0 0 degree a ! 0 00 such that if b 0 = c 0 = a then there exist B 2 b and C 2 c with B automorphic with C. However this attractive conjecture fails. To see this, Cooper =-=[1]-=- and independently Shore [12] constructed a jump class of noncappable degrees. In particular there is a \Sigma 2 set A ! T ; 00 such that for all recursively enumerable sets C, if C 0 j T A, then C ha... |

3 | Highness and bounding minimal pairs
- Downey, Lempp, et al.
- 1993
(Show Context)
Citation Context ...uestion of Downey and Stob concerning the distribution of hemimaximal degrees. These degree theoretical refinements use known techniques for making high 2 degrees adapted from Downey, Lempp and Shore =-=[2]-=- and methods from DowneyStob [5]. These techniques are taken wholesale and hence we do not give the whole construction but merely discuss the various strategies in detail. Notation is standard and fol... |

1 |
Jumps of hemimaximal sets,"Zeitschrift fur
- Downey, Stob
- 1991
(Show Context)
Citation Context ...t hemimaximal sets, halfs of nontrivial splittings of maximal sets, form an orbit that realizes many degrees including all high degrees, all jump classes and arbitrarily low degrees. (Downey and Stob =-=[3, 4]-=-) All of these theorems seem to point at the idea that the position of a recursively enumerable set in E always has ramifications in the degrees realized by its orbit. Such considerations lead to the ... |

1 |
and Mike Stob,"Minimal pairs in initial segments of the recursively enumerable degrees," submitted
- Downey
(Show Context)
Citation Context ... jump operator this result is nearly the best possible: ffl Certainly d cannot be high since there are hemimaximals below e which can be sent into any high recursively enumerable degree. (Downey-Stob =-=[5]-=-) ffl If d 1 were 0 then in the interval there would be hemimaximal sets that could be sent below e. The remaining possibility is that e could be high. We leave open the following, a positive solution... |

1 | There's no fat orbit, typewritten notes, (e-mail announcement - Harrington - 1991 |

1 | Definable properties of the lattice of recursively enumerable sets - Harrington, Soare |

1 |
Soare,"Automorphisms of the lattice of recursively enumerable sets, part 1 : maximal sets
- Robert
- 1974
(Show Context)
Citation Context ...sults here are the known results on orbits : the (Myhill-) Harrington proof that creative sets form a definable orbit and realize only sets of the complete m-degree (see Soare [14]), Soare's proof in =-=[13]-=- that the maximal sets form an orbit which realizes sets in each high degree by Martin [10], and the Downey-Stob proof that hemimaximal sets, halfs of nontrivial splittings of maximal sets, form an or... |

1 |
Soare,Recursively Enumerable Sets and Degrees
- Robert
- 1987
(Show Context)
Citation Context ... properties. Hallmark results here are the known results on orbits : the (Myhill-) Harrington proof that creative sets form a definable orbit and realize only sets of the complete m-degree (see Soare =-=[14]-=-), Soare's proof in [13] that the maximal sets form an orbit which realizes sets in each high degree by Martin [10], and the Downey-Stob proof that hemimaximal sets, halfs of nontrivial splittings of ... |

1 |
Soare and Michael Stob, "Relative recursive enumerability
- Robert
- 1982
(Show Context)
Citation Context ... clear if a weakened form of the false conjecture might work for double jump classes. (One cannot get this from our results since "double jump 4 inversion" does not work by the work of Soare=-= and Stob [15]-=-.) Thus we leave open the question below. Question. Is there an n such that for every n'th jump class Q there are degrees a and b such that for no A 2 a and B 2 b is A automorphic with B? One final co... |