## 1998], The Π3-theory of the computably enumerable Turing degrees is undecidable (1998)

Venue: | Trans. Amer. Math. Soc |

Citations: | 7 - 2 self |

### BibTeX

@ARTICLE{Lempp981998],the,

author = {Steffen Lempp and André Nies and Theodore and A. Slaman},

title = {1998], The Π3-theory of the computably enumerable Turing degrees is undecidable},

journal = {Trans. Amer. Math. Soc},

year = {1998},

volume = {350},

pages = {2719--2736}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We show the undecidability of the Π3-theory of the partial order of computably enumerable Turing degrees. Recursively enumerable (henceforth called computably enumerable) sets arise naturally in many areas of mathematics, for instance in the study of elementary theories, as solution sets of polynomials or as the word problems of finitely generated subgroups of finitely presented groups. Putting the computably enumerable sets

### Citations

117 |
Recursively enumerable sets of positive integers and their decision problems
- Post
- 1944
(Show Context)
Citation Context ...n). Here we are concerned with the partial order of Turing degrees of computably enumerable sets. This structure has been closely investigated for over fifty years, starting with Post’s seminal paper =-=[Po44]-=- and even before. Results of the 1950’s and early 1960’s, in particular the ground-breaking Sacks Density Theorem [Sa64], led Shoenfield [Sh65] to conjecture a strong homogeneity property (namely, tha... |

59 |
Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets (Springer-Verlag
- Soare
- 1987
(Show Context)
Citation Context ...n, respectively, for the computabilitytheoretic part of our result. For the computability-theoretic argument, we assume the reader to be familiar with 0 ′′′ -priority arguments. (Chapter XIV of Soare =-=[So87]-=- provides an introduction.) Our notation generally follows Soare [So87] with some exceptions: The names for the partial computable functionals used follow the “Chicago convention”, i.e., those built b... |

42 | Lower bounds for pairs of recursively enumerable degrees
- Lachlan
- 1966
(Show Context)
Citation Context ...ially supported by NSF Grant DMS-9500878. 2719 c○1998 American Mathematical Societys2720 S. LEMPP, A. NIES, AND T. A. SLAMAN Shoenfield’s conjecture was refuted by the minimal pair theorem of Lachlan =-=[La66]-=- and Yates [Ya66]. Further work in the 1970’s and 1980’s revealed more and more of the complexity of the poset of the computably enumerable degrees and led to a proof of the undecidability of its firs... |

35 |
The recursively enumerable degrees are dense
- Sacks
- 1964
(Show Context)
Citation Context ...n closely investigated for over fifty years, starting with Post’s seminal paper [Po44] and even before. Results of the 1950’s and early 1960’s, in particular the ground-breaking Sacks Density Theorem =-=[Sa64]-=-, led Shoenfield [Sh65] to conjecture a strong homogeneity property (namely, that any extension of embeddings of finite posets consistent with the theory of upper semilattices is always possible). Thi... |

16 |
The Undecidability of the Recursively Enumerable Degrees
- Harrington, Shelah
(Show Context)
Citation Context ...revealed more and more of the complexity of the poset of the computably enumerable degrees and led to a proof of the undecidability of its first-order theory by Harrington and Shelah, as announced in =-=[HS82]-=-. Once undecidability of a theory has been established, one reasonable next question is at which exact level of quantifier alternations (in brief: quantifier level), if at any, undecidability first oc... |

16 |
Undecidable fragments of elementary theories
- Nies
- 1996
(Show Context)
Citation Context ...ze quantifier alternations, using the undecidability of the Σ2-theory of the class of finite bipartite graphs (in the language of just one binary relation, without equality) and Nies’s Transfer Lemma =-=[Ni96]-=-. Our paper is organized as follows: In the next section, we present the statement of our theorem establishing our undecidability result and explain the algebraic part of the proof, i.e., the coding. ... |

11 |
A finite lattice without critical triple that cannot be embedded into the enumerable Turing degrees
- Lempp, Lerman
- 1997
(Show Context)
Citation Context ...ension of embeddings (of partial orderings) problem was shown to be decidable by Slaman and Soare [SSta]; the lattice embeddings problem, however, remains open (see Lerman [Le96] and Lempp and Lerman =-=[LLta]-=- for recent updates). On the undecidability side, by work of Harrington and Slaman, [HS82], the Π4-theory of the poset of the computably enumerable degrees was known to be undecidable. (A much easier ... |

7 |
A minimal pair of recursively enumerable degrees, J. Symbolic Logic 31
- Yates
- 1966
(Show Context)
Citation Context ...y NSF Grant DMS-9500878. 2719 c○1998 American Mathematical Societys2720 S. LEMPP, A. NIES, AND T. A. SLAMAN Shoenfield’s conjecture was refuted by the minimal pair theorem of Lachlan [La66] and Yates =-=[Ya66]-=-. Further work in the 1970’s and 1980’s revealed more and more of the complexity of the poset of the computably enumerable degrees and led to a proof of the undecidability of its first-order theory by... |

5 |
Degrees of unsolvability, Ann
- Sacks
- 1963
(Show Context)
Citation Context ... open interval (Π3). So the meaning of the question above is to determine which fragments of the theory experience has shown to be mathematically relevant are undecidable. By an early result of Sacks =-=[Sa63]-=-, the universal, or Π1-, fragment of the theory of the poset of the computably enumerable degrees is decidable since any existential statement consistent with the theory of partial orderings holds. Th... |

3 |
Embeddings into the recursively enumerable degrees, Computability, Enumerability, Unsolvability
- Lerman
- 1996
(Show Context)
Citation Context ...ave been considered. The extension of embeddings (of partial orderings) problem was shown to be decidable by Slaman and Soare [SSta]; the lattice embeddings problem, however, remains open (see Lerman =-=[Le96]-=- and Lempp and Lerman [LLta] for recent updates). On the undecidability side, by work of Harrington and Slaman, [HS82], the Π4-theory of the poset of the computably enumerable degrees was known to be ... |

3 |
Soare, Extension of embeddings in the computably enumerable degrees (to appear
- Slaman, I
- 2000
(Show Context)
Citation Context ...till remains an open problem. Two interesting fragments of the Π2-theory have been considered. The extension of embeddings (of partial orderings) problem was shown to be decidable by Slaman and Soare =-=[SSta]-=-; the lattice embeddings problem, however, remains open (see Lerman [Le96] and Lempp and Lerman [LLta] for recent updates). On the undecidability side, by work of Harrington and Slaman, [HS82], the Π4... |

2 |
Application of model theory to degrees of unsolvability, Symposium on the Theory of Models
- Shoenfield
- 1965
(Show Context)
Citation Context ...for over fifty years, starting with Post’s seminal paper [Po44] and even before. Results of the 1950’s and early 1960’s, in particular the ground-breaking Sacks Density Theorem [Sa64], led Shoenfield =-=[Sh65]-=- to conjecture a strong homogeneity property (namely, that any extension of embeddings of finite posets consistent with the theory of upper semilattices is always possible). This conjecture would have... |

1 |
Undecidability and one-types in the recursively enumerable degrees
- Ambos-Spies, Shore
- 1993
(Show Context)
Citation Context ...decidability side, by work of Harrington and Slaman, [HS82], the Π4-theory of the poset of the computably enumerable degrees was known to be undecidable. (A much easier proof by Ambos-Spies and Shore =-=[AS93]-=- gave the undecidability of the Π5-theory.) The present paper establishes undecidability for the Π3-theory by a very delicate coding so as to minimize quantifier alternations, using the undecidability... |

1 |
The undecidability of the Π4–theory of the r.e. wtt and Turing degrees
- Lempp, Nies
- 1995
(Show Context)
Citation Context ...rem 4, setting k =1and r = 2) to Theorem 2 in order to obtain the hereditary undecidability of the Π3theory of the computably enumerable degrees. A coding of finite bipartite graphs was first used in =-=[LNi95]-=- to establish the undecidability of the Π4-theory of the computably enumerable wtt–degrees. Here, we also use an ambiguous representation of vertices (as explained below) to ensure the coding is by Σ1... |