## Degree spectra of prime models (2004)

Venue: | J. Symbolic Logic |

Citations: | 8 - 2 self |

### BibTeX

@ARTICLE{Csima04degreespectra,

author = {Barbara F. Csima},

title = {Degree spectra of prime models},

journal = {J. Symbolic Logic},

year = {2004},

volume = {69},

pages = {430--442}

}

### OpenURL

### Abstract

2.1 Notation from model theory................... 4 2.2 F

### Citations

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Model theory: An introduction
- Marker
- 2002
(Show Context)
Citation Context ...theory T . Indeed, since the tree was built to have only Henkin paths, each path in [T ] will have Henkin witnesses. So the model can be obtained using the usual Henkin method, as described in Marker =-=[9]-=-. We are often interested in the atomic diagrams of models we are building. Given the effective enumeration f` i j i 2 !g of sentences in L c we get an effective sub-enumeration f` ij j j 2 !g of the ... |

38 |
Degrees coded in jumps of orderings
- Knight
- 1986
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Citation Context ...(A) = fdeg(D(B)) : B , = Ag dSp e (A) = fdeg(D e (B)) : B , = Ag We will often refer to the following result of Knight's, which tells us that the degree spectra are upward closed. Theorem 3.2 (Knight =-=[8]-=-). Let A be a countable structure in a relational language. Then either dSp a (A) is a singleton, or dSp a (A) is closed upwards. Corollary 3.3. For A such that dSp e (A) is not a singleton, (i) dSp e... |

24 |
Soare, Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets
- I
- 1987
(Show Context)
Citation Context ...es, we build a \Deltas0 2 approximation to a path on the tree, using the method of \Deltas0 2 permitting. 2 Notation and conventions 2.1 Notation from model theory We mostly use the notation of Soare =-=[14]-=- for computability theory, Chang and Keisler [1] for basic model theory, and the conventions of Harizanov [6] for computable model theory. We consider presentations A of models with universe !. For a ... |

21 |
Foundations of recursive model theory
- Millar
- 1978
(Show Context)
Citation Context ...t we have a complete decidable atomic theory, we can ask how complicated its prime model must be. Since the theory is complete and decidable, it has a decidable (and, hence, computable) model. Millar =-=[10]-=- ruled out the possibility of there always being a computable prime model by constructing a complete decidable atomic theory with no computable prime model. Denisov [4], Drobotun [5], and Millar [10] ... |

20 | Relative to any nonrecursive set
- Slaman
(Show Context)
Citation Context ...e spectrum of various structures can take has been widely studied. It was asked whether a structure could have the property that its degree spectrum was exactly all the non-computable degrees. Slaman =-=[13]-=- and Wehner [15] each constructed such countable first order structures, answering the question in the affirmative. The structures that they constructed were built particularly to answer the question,... |

17 |
Pure computable model theory, in Handbook of Recursive Mathematics
- Harizanov
- 1998
(Show Context)
Citation Context ...tation and conventions 2.1 Notation from model theory We mostly use the notation of Soare [14] for computability theory, Chang and Keisler [1] for basic model theory, and the conventions of Harizanov =-=[6]-=- for computable model theory. We consider presentations A of models with universe !. For a model A we write D(A) for the atomic diagram, and D e (A) for the elementary diagram. We say A is computable ... |

15 |
Enumerations, countable structures, and Turing degrees
- Wehner
- 1998
(Show Context)
Citation Context ...rious structures can take has been widely studied. It was asked whether a structure could have the property that its degree spectrum was exactly all the non-computable degrees. Slaman [13] and Wehner =-=[15]-=- each constructed such countable first order structures, answering the question in the affirmative. The structures that they constructed were built particularly to answer the question, and were not pr... |

12 |
Soare, Bounding prime models
- Csima, Hirschfeldt, et al.
(Show Context)
Citation Context ... low degree. That is to say, 8sthere is a low degree such that every complete atomic decidable theory has a prime model decidable in that degree. We call degrees with this property prime bounding. In =-=[3]-=-, Csima, Hirschfeldt, Knight, and Soare show that a \Deltas0 2 degree is prime bounding if and only if it is not low 2 . Thus there can be no low prime bounding degree, so we have a contradiction. Con... |

11 |
The ∆ 0 2-spectrum of a linear order
- Miller
(Show Context)
Citation Context ...on In the next two theorems, we will build models computable in particular \Deltas0 2 degrees. To do this we use the method of \Deltas0 2 permission. For a nice description of this method, see Miller =-=[12]-=-. Given a \Deltas0 2 set C, let fC s g s2! be a computable approximation of C. For s ? 0, let x s = maxfx j (9t ! s)[x ^ t ^ C s _ x = C t _ x]g t s = minft j x s ^ t ! s ^ C s _ x s = C t _ x s g: 13... |

9 |
Omitting types, type spectrums, and decidability
- Millar
(Show Context)
Citation Context ...ells us that for T a countable consistent theory, if f \Gammasj (x j ) g j2! is a countable family of partial types, all nonprincipal with respect to T , then T has a model omitting all \Gammasj . In =-=[11]-=-, Millar showed ways in which the classical theorem can and cannot be effectivized. Indeed, while he showed that given a complete decidable theory T , and a computable listing L of a subset \Psisof co... |

5 |
Homogeneous 0 ′ -elements in structural pre-orders
- Denisov
- 1989
(Show Context)
Citation Context ...nce, computable) model. Millar [10] ruled out the possibility of there always being a computable prime model by constructing a complete decidable atomic theory with no computable prime model. Denisov =-=[4]-=-, Drobotun [5], and Millar [10] showed that every complete decidable atomic theory has a prime model computable in ; 0 . In this paper we improve upon this result by showing: Theorem 1.1 (Prime Model ... |

5 |
Enumerations of simple models
- Drobotun
- 1977
(Show Context)
Citation Context ...e) model. Millar [10] ruled out the possibility of there always being a computable prime model by constructing a complete decidable atomic theory with no computable prime model. Denisov [4], Drobotun =-=[5]-=-, and Millar [10] showed that every complete decidable atomic theory has a prime model computable in ; 0 . In this paper we improve upon this result by showing: Theorem 1.1 (Prime Model Low Basis Theo... |

3 |
Model Theory, 3rd edn., Stud
- Chang, Keisler
- 1990
(Show Context)
Citation Context ... on the tree, using the method of \Deltas0 2 permitting. 2 Notation and conventions 2.1 Notation from model theory We mostly use the notation of Soare [14] for computability theory, Chang and Keisler =-=[1]-=- for basic model theory, and the conventions of Harizanov [6] for computable model theory. We consider presentations A of models with universe !. For a model A we write D(A) for the atomic diagram, an... |

2 |
A note on decidable model theory, Model Theory and Arithmetic
- Clote
- 1981
(Show Context)
Citation Context ...t, Knight, and Soare show that a \Deltas0 2 degree is prime bounding if and only if it is not low 2 . Thus there can be no low prime bounding degree, so we have a contradiction. Conjecture 4.2 (Clote =-=[2]-=-). There is a complete and decidable atomic theory T such that D e (A) * T 0 0 for every prime model A of T . Corollary 4.3. Clote's conjecture is false. Proof. By the prime model low basis theorem, e... |

2 |
Soare, \Pi classes and degrees of theories, Trans
- I
(Show Context)
Citation Context ...m does not follow immediately from the Low Basis Theorem. Indeed, suppose that one could always build a \Pis0 1 class of prime models of a complete decidable theory. By a result of Jockusch and Soare =-=[7]-=-, given the degree of any completion of Peano arithmetic and any \Pis0 1 class, there is a path in the class computable in the degree. Since there are completions of Peano arithmetic of low degree, th... |