## The theory of the degrees below 0 (1981)

Venue: | J. London Math. Soc |

Citations: | 18 - 6 self |

### BibTeX

@ARTICLE{Shore81thetheory,

author = {Richard A. Shore},

title = {The theory of the degrees below 0},

journal = {J. London Math. Soc},

year = {1981},

volume = {24},

pages = {1--14}

}

### Years of Citing Articles

### OpenURL

### Abstract

Degree theory, that is the study of the structure of the Turing degrees (or degrees of unsolvability) has been divided by Simpson [24; §5] into two parts—global and local. By the global theory he means the study of general structural properties of 3d— the degrees as a partially ordered set or uppersemilattice. The local theory concerns

### Citations

883 |
Theory of Recursive Functions and Effective Computability
- Rogers
- 1967
(Show Context)
Citation Context ... {x} E e x and {y} E e y. We will enumerate S recursively in E". Of course the key calculation is the Tarski-Kuratowski one showing that {</,j> | {/} E ^r {j} E } is If and so r.e. in E". (See Rogers =-=[19]-=- for such procedures.) Also note that given / and j with {/} £ and {/} £ total characteristic functions we can recursively find a k = g{i,j) such that {k} E = {i} E © {)} £ . We begin our enumeration ... |

131 | Borel determinacy - Martin - 1975 |

102 |
Trial and error predicates and solution to a problem of Mostowski
- Putnam
- 1965
(Show Context)
Citation Context ...x, s) =/(x) for all x I. Thus these functions have a \s-oo / certain air of effectiveness about them and the approximation procedures mimic real life learning processes. (See also Gold [5] and Putnam =-=[17]-=-.) One might well want to be a bit more generous in one's definition of local degree theory and include results about other small substructures of 3d such as <2>( ^ a) for a's other than 0' or ^[a, b]... |

95 |
Limiting recursion
- Gold
- 1965
(Show Context)
Citation Context ...nswers I lim g{x, s) =/(x) for all x I. Thus these functions have a \s-oo / certain air of effectiveness about them and the approximation procedures mimic real life learning processes. (See also Gold =-=[5]-=- and Putnam [17].) One might well want to be a bit more generous in one's definition of local degree theory and include results about other small substructures of 3d such as <2>( ^ a) for a's other th... |

48 |
On the degrees less than 0
- Sacks
- 1963
(Show Context)
Citation Context ... at the expense of considerable extra effort. Indeed Spector himself reopened the gap in the paper referred to above by proving that there is a minimal degree but only getting it below 0". Here Sacks =-=[20]-=- provides us with the local result by using a priority argument. Along these lines Lerman [11,12] has proved what is almost the analog of the complete global answer of Lachlan and Lebeuf [10] to the p... |

38 |
The upper semi-lattice of degrees of recursive unsolvability
- Kleene, Post
- 1954
(Show Context)
Citation Context ...ubstructures of 3d such as <2>( ^ a) for a's other than 0' or ^[a, b]. In any case local degree theory has its beginning in the same pathbreaking work that initiated the global theory—Kleene and Post =-=[8]-=-. In this paper Kleene and Post took explicit steps whenever they could to prove their structural results locally. Thus, the main embedding theorems (Sections 2 and 3) are all done between a and a' fo... |

38 |
On degrees of recursive unsolvability
- Spector
- 1956
(Show Context)
Citation Context ...and 3) are all done between a and a' for an arbitrary degree a. However, they were unable to prove one important structural result—that Q) is not a lattice—below 0'. This gap was filled in by Spector =-=[25]-=-. The pattern established here has continued. Structural results have usually been first proven in global forms and then pulled down below 0', generally at the expense of considerable extra effort. In... |

27 |
Minimal degrees and the jump operator
- Cooper
- 1973
(Show Context)
Citation Context ...dings below 0 (2) to ones below 0' and then to ones below arbitrary r.e. or high degrees. Thus for minimal degrees the corresponding results are due to Spector [25], Sacks [20], Yates [26] and Cooper =-=[1]-=- respectively. It is natural to ask what happens to the more general results on initial segments and their applications to undecidability questions.sTHE THEORY OF THE DEGREES BELOW 0' 1 1 Epstein [3] ... |

24 |
Countable initial segments of the degrees of unsolvability
- Lachlan, Lebeuf
- 1976
(Show Context)
Citation Context ...eting and the NSF for its support under Grant MCS 77-04013. Received 11 June, 1980. [J. LONDON MATH. SOC. (2), 24 (1981), 1-14]s2 RICHARD A. SHORE the relevant theories, globally of Th (9) by Lachlan =-=[9]-=- and locally of Th (9{ ^ 0')) by Lerman [11]. More elaborate initial segments results enabled Simpson [23] to show that Th{2>) is in fact recursively isomorphic to Th 2 (N, + , x), that is, full secon... |

22 |
Degrees of unsolvability: A survey of results
- Simpson
- 1977
(Show Context)
Citation Context ...nalogous investigation of Th {2>{ ^ 0')) to give the first proof that it is undecidable. In this paper we will prove the local version of Simpson's result and thereby verify a conjecture from Simpson =-=[24]-=-: Th (<3( ^ 0')) is recursively isomorphic to Th(N, +, x), that is, true first order arithmetic. Indeed, the same is true of Th (3)( ^ a)) for any arithmetic degree a ^ 0'. As corollaries to the proof... |

21 |
Double jumps of minimal degrees
- Posner
- 1978
(Show Context)
Citation Context ...initial segments of S>(^ a)(^[a,a']) for a ^ 0'. It says that some uppersemilattices of degree a (3) (a (4) ) are embeddable as initial segments of S>{ ^ a) (^[a, a']). However by Jockusch and Posner =-=[6]-=- the only initial segments which are lattices are a <2) (a (3) ) presentable. As there are of course lattices which are a (3) (a (4) ) but not a <2) (a (3) ) presentable there would seem to be no poss... |

16 |
First order theory of the degrees of recursive unsolvability
- Simpson
(Show Context)
Citation Context ... (2), 24 (1981), 1-14]s2 RICHARD A. SHORE the relevant theories, globally of Th (9) by Lachlan [9] and locally of Th (9{ ^ 0')) by Lerman [11]. More elaborate initial segments results enabled Simpson =-=[23]-=- to show that Th{2>) is in fact recursively isomorphic to Th 2 (N, + , x), that is, full second order arithmetic. Epstein [2,3] carried out enough of an analogous investigation of Th {2>{ ^ 0')) to gi... |

11 |
Reducibility orderings: theories, definability, and automorphisms
- Nerode, Shore
- 1980
(Show Context)
Citation Context ...'. As corollaries to the proof we will be able to distinguish between the theories of @[a, a'] for various degrees a. Our plan of attack is to basically follow the proofs in Nerode and Shore [14] and =-=[15]-=- that Th(^) is equivalent to second order arithmetic. Thus the idea is that by coding symmetric irreflexive binary relations by exact pairs for ideals in certain lattice initial segments of the degree... |

8 |
On homogeneity and definability in the first order theory of the Turing degrees
- Shore
- 1982
(Show Context)
Citation Context ... lattices in the style of Nerode and Shore [14] and [15] which makes the ideal generated by the degrees coding standard integers as easy to recover as possible. Essentially the one described in Shore =-=[22]-=- suffices although we need to make some additional properties of that lattice explicit to see that it works. We will briefly sketch the development of this coding apparatus. In Nerode and Shore [14] w... |

7 |
order logic and first order theories of reducibility orderings, The Kleene Symposium (Proc
- Nerode, Shore, et al.
- 1978
(Show Context)
Citation Context ...ree a ^ 0'. As corollaries to the proof we will be able to distinguish between the theories of @[a, a'] for various degrees a. Our plan of attack is to basically follow the proofs in Nerode and Shore =-=[14]-=- and [15] that Th(^) is equivalent to second order arithmetic. Thus the idea is that by coding symmetric irreflexive binary relations by exact pairs for ideals in certain lattice initial segments of t... |

4 |
Degrees of unsolvability of models
- Richter
- 1977
(Show Context)
Citation Context ...rove the main result on the theory of $){ ^ 0'), the stronger version gives improved results on homogeneity problems and precisely determines the degree of Q){ ^ a) for a ^ 0' in the sense of Richter =-=[18]-=-. We give its proof in this section and discuss applications in the next. LEMMA 2.1. Suppose we are given a set Eee and one W e Zf such that lw = {deg ({i} E ) | i € W] is an ideal in 2){ ^ e). (Note ... |

3 |
Initial segments of degrees below 0
- Epstein
- 1981
(Show Context)
Citation Context ...pectively. It is natural to ask what happens to the more general results on initial segments and their applications to undecidability questions.sTHE THEORY OF THE DEGREES BELOW 0' 1 1 Epstein [3] and =-=[4]-=- contain a lot of work along these lines. He embeds co+1 -chains below an arbitrary r.e. or high degree. By combining these results with various join theorems he manages to prove the undecidability of... |

2 |
Initial segments of the degrees below 0
- Lerman
- 1978
(Show Context)
Citation Context ...aper referred to above by proving that there is a minimal degree but only getting it below 0". Here Sacks [20] provides us with the local result by using a priority argument. Along these lines Lerman =-=[11,12]-=- has proved what is almost the analog of the complete global answer of Lachlan and Lebeuf [10] to the problem of classifying the countable initial segments of 3d\ every 0" presentable lattice can be e... |

2 |
Degrees of Unsolvability, Springer-Verlag
- Lerman
- 1983
(Show Context)
Citation Context ...aper referred to above by proving that there is a minimal degree but only getting it below 0". Here Sacks [20] provides us with the local result by using a priority argument. Along these lines Lerman =-=[11,12]-=- has proved what is almost the analog of the complete global answer of Lachlan and Lebeuf [10] to the problem of classifying the countable initial segments of 3d\ every 0" presentable lattice can be e... |

1 |
Analysis and degrees ofunsolvability ^ 0
- Epstein
- 1978
(Show Context)
Citation Context ...) by Lerman [11]. More elaborate initial segments results enabled Simpson [23] to show that Th{2>) is in fact recursively isomorphic to Th 2 (N, + , x), that is, full second order arithmetic. Epstein =-=[2,3]-=- carried out enough of an analogous investigation of Th {2>{ ^ 0')) to give the first proof that it is undecidable. In this paper we will prove the local version of Simpson's result and thereby verify... |

1 |
Comsat General Integrated Systems, Super-Filsyn Users Manual
- unknown authors
- 1982
(Show Context)
Citation Context ...) by Lerman [11]. More elaborate initial segments results enabled Simpson [23] to show that Th{2>) is in fact recursively isomorphic to Th 2 (N, + , x), that is, full second order arithmetic. Epstein =-=[2,3]-=- carried out enough of an analogous investigation of Th {2>{ ^ 0')) to give the first proof that it is undecidable. In this paper we will prove the local version of Simpson's result and thereby verify... |

1 |
Solovay, "Fixed points of jump preserving automorphisms of degrees
- M
- 1977
(Show Context)
Citation Context ...the results to 0'. If one restricts ones attention to degrees above 0 (2) then one can use the results of Lachlan and Lebeuf [10] and the fact that their embeddings give e" = 0" (Jockusch and Solovay =-=[7]-=-) instead of Lerman [12] to derive all the results of this section, that is, Theorems 3.2, 3.4 and 3.6 can all be derived in this way for a, b ^ 0 (2) . 4. Working below r.e. or high degrees The trend... |

1 |
Countable initial segments of the degrees ofunsolvability
- Lachlan, Lebeuf
- 1976
(Show Context)
Citation Context ...re Sacks [20] provides us with the local result by using a priority argument. Along these lines Lerman [11,12] has proved what is almost the analog of the complete global answer of Lachlan and Lebeuf =-=[10]-=- to the problem of classifying the countable initial segments of 3d\ every 0" presentable lattice can be embedded as an intial segment of 3d{ ^ 0'). Again priority arguments and approximations play a ... |

1 | The degrees below 0 - Posner - 1979 |

1 | Degrees of unsolvability (North-Holland/American - Shoenfield - 1971 |

1 |
Initial segments of the degrees of unsolvability II: minimal degrees
- Yates
- 1970
(Show Context)
Citation Context ...to go from embeddings below 0 (2) to ones below 0' and then to ones below arbitrary r.e. or high degrees. Thus for minimal degrees the corresponding results are due to Spector [25], Sacks [20], Yates =-=[26]-=- and Cooper [1] respectively. It is natural to ask what happens to the more general results on initial segments and their applications to undecidability questions.sTHE THEORY OF THE DEGREES BELOW 0' 1... |