## Almost everywhere domination

Venue: | J. Symbolic Logic |

Citations: | 34 - 16 self |

### BibTeX

@ARTICLE{Dobrinen_almosteverywhere,

author = {Natasha L. Dobrinen and Stephen G. Simpson},

title = {Almost everywhere domination},

journal = {J. Symbolic Logic},

year = {},

volume = {69},

pages = {914--922}

}

### OpenURL

### Abstract

ATuringdegreea is said to be almost everywhere dominating if, for almost all X ∈ 2 ω with respect to the “fair coin ” probability measure on 2 ω,andforallg: ω → ω Turing reducible to X, thereexistsf: ω → ω of Turing degree a which dominates g. We study the problem of characterizing the almost everywhere dominating Turing degrees and other, similarly defined classes of Turing degrees. We relate this problem to some questions in the reverse mathematics of measure theory. 1

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Citation Context ...ining the weakest set existence axioms needed to prove specific mathematical theorems. This is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson =-=[12]-=-. Other results on the reverse mathematics of 6smeasure theory are in the papers of Yu [14, 15, 16, 17, 18], Yu/Simpson [19], and Brown/Giusto/Simpson [1]. A well known result in measure theory assert... |

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Citation Context ...ecursive functions, is not almost everywhere dominating. In this paper we raise the problem of characterizing the Turing degrees which are almost everywhere dominating. The following theorem of Kurtz =-=[5]-=- implies that 0 ′ , the Turing degree of the Halting Problem, is almost everywhere dominating. We consider an apparently more restrictive property. Definition 2.2. We say that A ∈ 2ω is almost everywh... |

55 |
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Citation Context ...ee. The following are pairwise equivalent. 1. a is almost everywhere dominating. 2. a is uniformly almost everywhere dominating. 3. a ′ ≥ 0 ′′ . Toward Conjecture 2.9, the following theorem of Martin =-=[6]-=- is well known. Say that A is uniformly dominating if there exists f ∈ REC[A] such that f dominates every g ∈ REC. Again, this is a property of the Turing degree of A. Theorem 2.10 (Martin [6]). ATuri... |

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Citation Context ...18] (a consequence of the Low Basis Theorem) that there exists an ù-model of WKL0 in which (?) fails. We thank the referee for suggesting this observation. Furthermore, using Theorem III.2.1 of Kautz =-=[4]-=-, we can build anù-modelM ofWKL0 such that (8X 2M ) (9Y 2M ) (Y isù-random relative to X ), and (8X 2 M ) (9Y 2 M ) (Y is ù-generic relative to X ), yet (8Y 2M ) (Y 0 6T 000), hence (?) fails inM . R... |

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Citation Context ...d subset of 2ω , hence for each n, � � � X {e} (n) � X ∈ Pe,i is finite, by compactness of 2 ω .Thuswehave ∀e ∀i ∀n ∃m ∀X (P(〈0,...,0, 1〉 � �� � e � X,i) ⇒{e} X (n) ≤ m). Now, by Lemma 3.5 of Simpson =-=[11]-=- relativized to A, the predicate ⎛ ∀X ⎝P(〈0,...,0, 1〉 � �� � e � X,i) ⇒{e} X ⎞ (n) ≤ m⎠ is Σ 0,A 1 . Hence by Σ 0,A 1 uniformization we find g : ω × ω × ω → ω recursive in A such that ∀e ∀X ∀i (P(〈0,.... |

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Citation Context ...ermining the weakest set existence axiomsneeded to prove speciÞcmathematical theorems. This is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson =-=[12]-=-. Other results on the reverse mathematics of measure theory are in the papers of Yu [14, 15, 16, 17, 18], Yu/Simpson [19], and Brown/Giusto/Simpson [1]. ALMOST EVERYWHERE DOMINATION 919 A well known ... |

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Citation Context ... is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of 6smeasure theory are in the papers of Yu =-=[14, 15, 16, 17, 18]-=-, Yu/Simpson [19], and Brown/Giusto/Simpson [1]. A well known result in measure theory asserts that the fair coin measure µ is regular. This means that measurable sets are approximable from within by ... |

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Citation Context ... is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of 6smeasure theory are in the papers of Yu =-=[14, 15, 16, 17, 18]-=-, Yu/Simpson [19], and Brown/Giusto/Simpson [1]. A well known result in measure theory asserts that the fair coin measure µ is regular. This means that measurable sets are approximable from within by ... |

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Citation Context ... subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of 6smeasure theory are in the papers of Yu [14, 15, 16, 17, 18], Yu/Simpson =-=[19]-=-, and Brown/Giusto/Simpson [1]. A well known result in measure theory asserts that the fair coin measure µ is regular. This means that measurable sets are approximable from within by Fσ sets and from ... |

4 |
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Citation Context ... is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of 6smeasure theory are in the papers of Yu =-=[14, 15, 16, 17, 18]-=-, Yu/Simpson [19], and Brown/Giusto/Simpson [1]. A well known result in measure theory asserts that the fair coin measure µ is regular. This means that measurable sets are approximable from within by ... |

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Citation Context ...any open sets. Regularity of ì means: For every measurable set Q 2ù there exist an Fó set S and a Gä set P such that S Q P and ì(S) = ì(Q) = ì(P). See for example the classic textbook of Halmos =-=[2]-=-. Attempting to reverse this measure-theoretic result, we encounter the diculty that arbitrary measurable sets cannot be discussed in the language of second order arithmetic. However, we candiscuss s... |

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Citation Context ....18] (a consequence of the Low Basis Theorem) that there exists an !-model of WKL0 in which (?) fails. We thank the refereefor suggesting this observation. Furthermore, using Theorem III.2.1 of Kautz =-=[4]-=-, we can build an !-model M of WKL0 such that (8X2M )(9Y 2M )(Y is !-random relative to X), and (8X2M )(9Y 2M )(Y is !-generic relative to X), yet(8 Y 2M )(Y 0 6>=T 000), hence (?) fails in M . Refere... |

2 |
theorem is equivalent to arithmetical comprehension
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Citation Context ...is is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of measure theory are in the papers of Yu =-=[14, 15, 16, 17, 18]-=-, Yu/Simpson [19], and Brown/Giusto/Simpson [1]. ALMOST EVERYWHERE DOMINATION 919 A well known result in measure theory asserts that the fair coin measure ì is regular. This means that measurable sets... |

2 |
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Citation Context ...is is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of measure theory are in the papers of Yu =-=[14, 15, 16, 17, 18]-=-, Yu/Simpson [19], and Brown/Giusto/Simpson [1]. ALMOST EVERYWHERE DOMINATION 919 A well known result in measure theory asserts that the fair coin measure ì is regular. This means that measurable sets... |

1 |
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Citation Context ....1, it would be natural to conjecture that for almost all X ∈ 2ω and all g ∈ REC[X] thereexistsf∈REC such that f dominates g. However, this is not the case, as shown by the following result of Martin =-=[7]-=-. Since the proof of Theorem 1.2 has not been published, we present it below. Theorem 1.2 (Martin [7]). For almost all X ∈ 2ω there exists g ∈ REC[X] such that g is not dominated by any f ∈ REC. Proof... |

1 |
A.Martin,Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift f¬ur Mathematische Logik und Grundlagen der
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Citation Context ...egree. The following are pairwise equivalent. 1. a is almost everywhere dominating. 2. a is uniformly almost everywhere dominating. 3. a0 000. Toward Conjecture 2.9, the following theorem of Martin =-=[6]-=- is well known. Say that A is uniformly dominating if there exists f 2 REC[A] such that f dominates every g 2 REC. Again, this is a property of the Turing degree of A. Theorem 2.10 (Martin [6]). ATuri... |

1 |
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Citation Context ...t would be natural to conjecture that for almost all X 2 2ù and all g 2 REC[X ] there exists f 2 REC such that f dominates g. However, this is not the case, as shown by the following result of Martin =-=[7]-=-. Since the proof of Theorem 1.2 has not been published, we present it below. Theorem 1.2 (Martin [7]). For almost all X 2 2ù there exists g 2 REC[X ] such that g is not dominated by any f 2 REC. Proo... |

1 |
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(Show Context)
Citation Context ...nals belonging toM . It is known that, for almost all X 2 2ù, M [X ] is a model of ZFC. This leads to a forcing-free proof of the independence of the Continuum Hypothesis. See the exposition of Sacks =-=[8]-=-. The purpose of this paper is to investigate recursion-theoretic analogs of Theorem 1.1, replacing the set-theoretic ground modelM by the recursion-theoretic ground model REC = ff 2 ùù j f is recursi... |

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(Show Context)
Citation Context ...d subset of 2ù , hence for each n, fegX (n)sX 2 Pe;i is Þnite, by compactness of 2ù . Thus we have 8e 8i 8n 9m 8X P(h0; : : : ; 0| {z } e ; 1iaX; i)=)fegX (n) m : Now, by Lemma 3.5 of Simpson =-=[11]-=- relativized to A, the predicate 8X P(h0; : : : ; 0| {z } e ; 1iaX; i)=)fegX (n) m is ¶0;A1 . Hence by ¶ 0;A 1 uniformization we Þnd g : ù ù ù ! ù recursive in A such that 8e 8X 8i P(h0; :... |

1 |
convergence theorems and reverse mathematics,Mathematical Logic Quarterly
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Citation Context ...is is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of measure theory are in the papers of Yu =-=[14, 15, 16, 17, 18]-=-, Yu/Simpson [19], and Brown/Giusto/Simpson [1]. ALMOST EVERYWHERE DOMINATION 919 A well known result in measure theory asserts that the fair coin measure ì is regular. This means that measurable sets... |

1 |
Measure theory and weak K¬onigÕs lemma, Archive for
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(Show Context)
Citation Context ...of subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of measure theory are in the papers of Yu [14, 15, 16, 17, 18], Yu/Simpson =-=[19]-=-, and Brown/Giusto/Simpson [1]. ALMOST EVERYWHERE DOMINATION 919 A well known result in measure theory asserts that the fair coin measure ì is regular. This means that measurable sets are approximable... |