## A General Framework for Priority Arguments (1995)

Venue: | THE BULLETIN OF SYMBOLIC LOGIC |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Lempp95ageneral,

author = {Steffen Lempp and Manuel Lerman},

title = {A General Framework for Priority Arguments},

journal = {THE BULLETIN OF SYMBOLIC LOGIC},

year = {1995},

volume = {1},

pages = {189--201}

}

### OpenURL

### Abstract

### Citations

119 |
Borel determinacy
- Martin
- 1975
(Show Context)
Citation Context ...rington, and many others; and in e#ective model theory byNerode, Remmel, Ash, Knight, and others. There have been several important applications outside computability theory. Among these are Martin's =-=[18]-=- original proof of the Axiom of Borel Determinacy, Solovay's [27] characterization of the degrees of models of true arithmetic, and Slaman's result that ACA 0 is not conservative over RCA 0 +Ramsey's ... |

117 |
Recursively enumerable sets of positive integers and their decision problems
- Post
- 1944
(Show Context)
Citation Context ...bolic Logic Volume 1, Number 2, June 1995 A GENERAL FRAMEWORK FOR PRIORITY ARGUMENTS STEFFEN LEMPP AND MANUEL LERMAN The degrees of unsolvability were introduced in the ground-breaking papers of Post =-=[20]-=- and Kleene and Post [7] as an attempt to measure the information content of sets of natural numbers. Kleene and Post were interested in the relative complexity of decision problems arising naturally ... |

72 |
Soare, Recursively Enumerable Sets and Degrees
- I
- 1987
(Show Context)
Citation Context ... framework which we have developed [15], [16] is built on the tree of strategies approach to priority arguments which was introduced by Lachlan [14], developed by Harrington, and popularized by Soare =-=[26]-=-. However, we use a sequence of trees of strategies rather than a single tree. We begin by assigning requirements of high quantifier complexity to a tree T n of level n for some n. We then require a l... |

62 |
Degrees of Unsolvability
- Sacks
- 1963
(Show Context)
Citation Context ...h to generate action, and to guess at the truth of the full sentence. At the next level of quantifier complexity, the corresponding method has been called infinite injury, and was discovered by Sacks =-=[21]-=-, Shoenfield [22], and Yates [28] and developed primarily by Sacks. The level after that was initially called monstrous injury, and was discovered by Lachlan [14]. Harrington introduced a nicer classi... |

47 |
Two recursively enumerable sets of incomparable degrees of unsolvability
- Friedberg
- 1957
(Show Context)
Citation Context ...enumerable degrees have a least and greatest element, and asked whether there were other enumerable degrees. This problem, which became known as Post's Problem, was solved a decade later by Friedberg =-=[5]-=- and Mu cnik [19], and their solutions introduced a new technique, the priority method, which is the subject of this paper. Received August 17, 1994; revised December 1, 1994. The first author's resea... |

37 |
The upper semi-lattice of degrees of recursive unsolvability
- Kleene, Post
- 1954
(Show Context)
Citation Context ...ber 2, June 1995 A GENERAL FRAMEWORK FOR PRIORITY ARGUMENTS STEFFEN LEMPP AND MANUEL LERMAN The degrees of unsolvability were introduced in the ground-breaking papers of Post [20] and Kleene and Post =-=[7]-=- as an attempt to measure the information content of sets of natural numbers. Kleene and Post were interested in the relative complexity of decision problems arising naturally in mathematics; in parti... |

17 |
On some games which are relevant to the theory of recursively enumerable sets
- Lachlan
- 1970
(Show Context)
Citation Context ...undancy in proofs using such arguments. Sacks [21] provided such a framework for finite injury, and Lachlan made some early attempts for infinite injury as well, taking both a game-theoretic approach =-=[12]-=-, and an e#ective Baire category approach [13]. Yates [29] developed an approach for combining infinite injury arguments with e#ective perfect closed set forcing in the setting of the degrees below 0 ... |

13 |
On the degrees of index sets
- Yates
- 1966
(Show Context)
Citation Context ...s at the truth of the full sentence. At the next level of quantifier complexity, the corresponding method has been called infinite injury, and was discovered by Sacks [21], Shoenfield [22], and Yates =-=[28]-=- and developed primarily by Sacks. The level after that was initially called monstrous injury, and was discovered by Lachlan [14]. Harrington introduced a nicer classification of the levels of complex... |

9 |
Stability of recursive structures in arithmetical degrees
- Ash
- 1986
(Show Context)
Citation Context ...to find a general way to approach priority arguments at all arithmetical levels, and accomplished this by combining the tree of strategies approach with the use of the Kleene Fixed-Point Theorem. Ash =-=[1]-=-, [2], and Knight [8] have developed various frameworks which apply to restricted classes of problems in e#ective model theory using trees of enumerations, but cover all hyperarithmetic levels of comp... |

9 |
The decidability of the existential theory of the poset of the recursively enumerable degrees with jump relations
- Lempp, Lerman
- 1996
(Show Context)
Citation Context ... Another approach, using trees of trees, has been introduced by Groszek and Slaman [6]. The latter two approaches have influenced the development of our framework, which is substantially developed in =-=[15]-=-, and will be fully developed in [16]. This framework takes an inductive approach and uses a separate tree of strategies for each level of the induction. While it has been developed only for finite le... |

8 |
Rigidity and definability in the noncomputable universe
- Cooper
- 1994
(Show Context)
Citation Context ...tructure, i.e., is it the case that the degrees have no non-trivial automorphisms? Many people have worked on this problem, and combined results of Shore [24], Slaman and Woodin [25], and Cooper [4], =-=[3]-=-, show that there are few, if any, non-trivial automorphisms, and any such automorphism must move degrees to nearby degrees. Other research has pursued answers to similar questions about substructures... |

4 |
A Metatheorem for Construction by Finitely Many Workers
- Knight
- 1990
(Show Context)
Citation Context ... to approach priority arguments at all arithmetical levels, and accomplished this by combining the tree of strategies approach with the use of the Kleene Fixed-Point Theorem. Ash [1], [2], and Knight =-=[8]-=- have developed various frameworks which apply to restricted classes of problems in e#ective model theory using trees of enumerations, but cover all hyperarithmetic levels of complexity of the priorit... |

3 |
Undecidable and creative theories
- Shoenfield
- 1961
(Show Context)
Citation Context ...ion, and to guess at the truth of the full sentence. At the next level of quantifier complexity, the corresponding method has been called infinite injury, and was discovered by Sacks [21], Shoenfield =-=[22]-=-, and Yates [28] and developed primarily by Sacks. The level after that was initially called monstrous injury, and was discovered by Lachlan [14]. Harrington introduced a nicer classification of the l... |

3 |
On homogeneity and definability in the first-order theory of the Turing degrees
- Shore
- 1982
(Show Context)
Citation Context ...se to degrees which look di#erent inside the structure, i.e., is it the case that the degrees have no non-trivial automorphisms? Many people have worked on this problem, and combined results of Shore =-=[24]-=-, Slaman and Woodin [25], and Cooper [4], [3], show that there are few, if any, non-trivial automorphisms, and any such automorphism must move degrees to nearby degrees. Other research has pursued ans... |

2 |
Degrees of models of true arithmetic, preliminary version, unpublished manuscript
- Solovay
- 1984
(Show Context)
Citation Context ...Remmel, Ash, Knight, and others. There have been several important applications outside computability theory. Among these are Martin's [18] original proof of the Axiom of Borel Determinacy, Solovay's =-=[27]-=- characterization of the degrees of models of true arithmetic, and Slaman's result that ACA 0 is not conservative over RCA 0 +Ramsey's Theorem for Pairs. It is our hope that the framework which we pre... |

2 |
comeagre sets, and degrees of unsolvability
- games
- 1976
(Show Context)
Citation Context ...d such a framework for finite injury, and Lachlan made some early attempts for infinite injury as well, taking both a game-theoretic approach [12], and an e#ective Baire category approach [13]. Yates =-=[29]-=- developed an approach for combining infinite injury arguments with e#ective perfect closed set forcing in the setting of the degrees below 0 # , using Banach-Mazur games to model the constructions. S... |

1 |
Topological framework for infinite injury
- Kontostathis
(Show Context)
Citation Context ...s to model the constructions. Subsequent frameworks for priority arguments of restricted complexity have been developed by Shoenfield [23] using a tree of strategies, and Kontostathis [10], [11], and =-=[9]-=- using an e#ective Baire category approach. Harrington was the first to find a general way to approach priority arguments at all arithmetical levels, and accomplished this by combining the tree of str... |

1 |
framework for non-priority, Zeitschrift f ur Mathematische Logik und Grundlagen der
- Topological
- 1991
(Show Context)
Citation Context ...anach-Mazur games to model the constructions. Subsequent frameworks for priority arguments of restricted complexity have been developed by Shoenfield [23] using a tree of strategies, and Kontostathis =-=[10]-=-, [11], and [9] using an e#ective Baire category approach. Harrington was the first to find a general way to approach priority arguments at all arithmetical levels, and accomplished this by combining ... |

1 |
framework for finite injury, Zeitschrift f ur Mathematische Logik und Grundlagen der
- Topological
- 1992
(Show Context)
Citation Context ...Mazur games to model the constructions. Subsequent frameworks for priority arguments of restricted complexity have been developed by Shoenfield [23] using a tree of strategies, and Kontostathis [10], =-=[11]-=-, and [9] using an e#ective Baire category approach. Harrington was the first to find a general way to approach priority arguments at all arithmetical levels, and accomplished this by combining the tr... |

1 |
Minimal pair constructions and iterated trees of strategies, Logical methods, in honor of Anil Nerode's 60th birthday, Birkh auser
- Lempp, Lerman, et al.
- 1993
(Show Context)
Citation Context ...alized in one step. (An advantage of the framework in the latter case is that its presentation is as close as possible to the standard presentation, if an inductive component is to be introduced.) In =-=[17]-=-, we used the framework to prove a new theorem, the existence of a minimal pair of enumerable degrees whose jumps form a minimal pair over 0 # . This theorem can probably be generalized to carry the s... |

1 |
cnik, On the unsolvability of the problem of reducibility in the theory of algorithms
- Mu
- 1956
(Show Context)
Citation Context ...es have a least and greatest element, and asked whether there were other enumerable degrees. This problem, which became known as Post's Problem, was solved a decade later by Friedberg [5] and Mu cnik =-=[19]-=-, and their solutions introduced a new technique, the priority method, which is the subject of this paper. Received August 17, 1994; revised December 1, 1994. The first author's research was partially... |