## Mass problems and measure-theoretic regularity (2009)

Citations: | 4 - 3 self |

### BibTeX

@MISC{Simpson09massproblems,

author = {Stephen G. Simpson},

title = {Mass problems and measure-theoretic regularity},

year = {2009}

}

### OpenURL

### Abstract

Research supported by NSF grants DMS-0600823 and DMS-0652637.

### Citations

1280 |
On computable numbers, with an application to the Entscheidungsproblem
- Turing
- 1936
(Show Context)
Citation Context ...s, and measure-theoretic regularity. Degrees of unsolvability Degrees of unsolvability are a well known and highly developed research area which grew out of a fundamental discovery due to Turing 1936 =-=[54]-=-: the halting problem for Turing machines is algorithmically unsolvable. As is well known, Turing’s example of an unsolvable mathematical problem was the first such example, and as such it revolutioni... |

875 |
Theory of Recursive Functions and Effective Computability
- Rogers
- 1967
(Show Context)
Citation Context ...er research during the period 1960–1990 was motivated by structural and methodological questions concerning various degree structures. Some classical treatises from this period are Sacks [33], Rogers =-=[32]-=-, Shoenfield [36], Lerman [22], Soare [52], Odifreddi [29, 30]. 2The classical theory of degrees of unsolvability was concerned mainly with decision problems and their Turing degrees. A more recent t... |

754 | Introduction to Metamathematics - Kleene - 1950 |

500 |
Recursively enumerable sets and degrees
- Soare
- 1978
(Show Context)
Citation Context ...s motivated by structural and methodological questions concerning various degree structures. Some classical treatises from this period are Sacks [33], Rogers [32], Shoenfield [36], Lerman [22], Soare =-=[52]-=-, Odifreddi [29, 30]. 2The classical theory of degrees of unsolvability was concerned mainly with decision problems and their Turing degrees. A more recent trend has been to focus instead on mass pro... |

342 |
The definition of random sequences
- Martin-Löf
- 1966
(Show Context)
Citation Context ... including a version of countable additivity and a version of the Vitali Covering Lemma. Subsequently it was noticed that WWKL0 is closely related to algorithmic randomness in the sense of Martin-Löf =-=[23, 28, 11, 50]-=-. Indeed, the principal axiom of WWKL0 turned out to be equivalent over RCA0 to the statement ∀X ∃Y (Y is Martin-Löf random relative to X). See also our summary in [40, 49, Section X.1]. Later Simpson... |

219 | Mathematical logic - Shoenfield - 1967 |

208 |
Subsystems of Second Order Arithmetic
- Simpson
- 1999
(Show Context)
Citation Context ...ed in certain formal, deductive systems. The most important formal systems for reverse mathematics are the so-called “Big Five”: RCA0, WKL0, ACA0, ATR0, Π 1 1 -CA0, corresponding to Chapters II–VI of =-=[40, 49]-=-. The standard reference for reverse mathematics is Simpson [40, 49]. See also our recent survey in [47]. The present paper includes a contribution to the reverse mathematics of measure theory. In ord... |

171 |
Algorithmic Randomness and Complexity
- Downey, Hirshfeldt
(Show Context)
Citation Context ... including a version of countable additivity and a version of the Vitali Covering Lemma. Subsequently it was noticed that WWKL0 is closely related to algorithmic randomness in the sense of Martin-Löf =-=[23, 28, 11, 50]-=-. Indeed, the principal axiom of WWKL0 turned out to be equivalent over RCA0 to the statement ∀X ∃Y (Y is Martin-Löf random relative to X). See also our summary in [40, 49, Section X.1]. Later Simpson... |

121 |
Recursively enumerable sets of positive integers and their decision problems
- POST
- 1944
(Show Context)
Citation Context ... of unsolvable mathematical problems and to quantify their unsolvability by classifying them according to the “amount” or degree of unsolvability which is inherent in them. See for instance Post 1944 =-=[31]-=- and Kleene/Post 1954 [20]. Later research during the period 1960–1990 was motivated by structural and methodological questions concerning various degree structures. Some classical treatises from this... |

113 | Computability and Randomness
- Nies
(Show Context)
Citation Context ... including a version of countable additivity and a version of the Vitali Covering Lemma. Subsequently it was noticed that WWKL0 is closely related to algorithmic randomness in the sense of Martin-Löf =-=[23, 28, 11, 50]-=-. Indeed, the principal axiom of WWKL0 turned out to be equivalent over RCA0 to the statement ∀X ∃Y (Y is Martin-Löf random relative to X). See also our summary in [40, 49, Section X.1]. Later Simpson... |

84 | Lowness properties and randomness - Nies |

80 |
Degrees of Unsolvability
- Lerman
- 1983
(Show Context)
Citation Context ...1960–1990 was motivated by structural and methodological questions concerning various degree structures. Some classical treatises from this period are Sacks [33], Rogers [32], Shoenfield [36], Lerman =-=[22]-=-, Soare [52], Odifreddi [29, 30]. 2The classical theory of degrees of unsolvability was concerned mainly with decision problems and their Turing degrees. A more recent trend has been to focus instead... |

61 |
Computable Structures and the Hyperarithmetical Hierarchy. Number 144
- Ash, Knight
(Show Context)
Citation Context ... ≤T X ⊕ Y ′ . Proof. This follows from Theorem 4.7 plus the well known fact that X (α) is a Σ 0,X 3 singleton. For a proof of this fact, see any textbook of hyperarithmetical theory, e.g., Ash/Knight =-=[1]-=-, Rogers [32, Chapter 16], Sacks [34, Part A], Shoenfield [35, Sections 7.8–7.11], Simpson [40, 49, Section VIII.3]. 11Corollary 4.10. For each α < ω CK 1 , 0 (α) ≤LR Y implies 0 (α+1) ≤T Y ′ . Proof... |

46 | Degrees of Random Sets - Kautz - 1991 |

38 |
The upper semi-lattice of degrees of recursive unsolvability, Annals of Mathematics 59
- Kleene, Post
- 1954
(Show Context)
Citation Context ...l problems and to quantify their unsolvability by classifying them according to the “amount” or degree of unsolvability which is inherent in them. See for instance Post 1944 [31] and Kleene/Post 1954 =-=[20]-=-. Later research during the period 1960–1990 was motivated by structural and methodological questions concerning various degree structures. Some classical treatises from this period are Sacks [33], Ro... |

36 | Mass problems and randomness
- Simpson
- 2005
(Show Context)
Citation Context ...itionism, algorithmic randomness, Kolmogorov complexity, resource bounded computational complexity, subrecursive hierarchies, and unsolvable mathematical problems. See in particular our recent papers =-=[9, 41, 43, 44, 45, 46, 48]-=- and our forthcoming treatise [51]. We emphasize that the study of mass problems offers a path along which the study of degrees of unsolvability can return to and reconnect with its roots in the found... |

34 | A splitting theorem for the Medvedev and Muchnik lattices
- Binns
(Show Context)
Citation Context ...ev [24], and Muchnik [25] concerning the so-called “intuitionistic calculus of problems” [21]. In particular Muchnik [25] showed that Dw is a complete Brouwerian lattice. Some recent papers on Ew are =-=[2, 4, 9, 41, 43, 44, 45, 46]-=-. In these papers Ew is often denoted Pw. Lemma 6.4 below implies that Ew includes a large and significant part of Dw. For additional historical references see [45]. In this paper we study certain mas... |

34 | Almost everywhere domination
- Dobrinen, Simpson
(Show Context)
Citation Context ...Martin-Löf random}) 4was the first example of a specific, natural degree in Ew other than 0 and 1. The second wave dealing with measure-theoretic regularity was initiated in 2002 by Dobrinen/Simpson =-=[10]-=- and was centered around our notion of almost everywhere domination. In [10] we defined a Turing oracle Y to be almost everywhere dominating if for all Turing oracles X except a set of measure 0, ever... |

29 | Mass problems and hyperarithmeticity
- Cole, Simpson
(Show Context)
Citation Context ...itionism, algorithmic randomness, Kolmogorov complexity, resource bounded computational complexity, subrecursive hierarchies, and unsolvable mathematical problems. See in particular our recent papers =-=[9, 41, 43, 44, 45, 46, 48]-=- and our forthcoming treatise [51]. We emphasize that the study of mass problems offers a path along which the study of degrees of unsolvability can return to and reconnect with its roots in the found... |

29 | Reals which compute little
- Nies
- 2006
(Show Context)
Citation Context ...we can find a Y -r.e. set J such that µ(J) < ∞ and I ⊆ J. Proof. This result is due to Kjos-Hanssen/Miller/Solomon [18]. See also our exposition in [42]. The next definition and lemma are due to Nies =-=[27]-=- and Simpson [42, Definition 8.3, Lemma 8.4]. Definition 4.4 (jump-traceability). Recall Definition 2.6 where we defined the Turing jump operator X ↦→ X ′ in a somewhat unusual manner. We say that X i... |

28 | Uniform almost everywhere domination
- Cholak, Greenberg, et al.
(Show Context)
Citation Context ...set of the same measure, if and only if 0 ′ ≤LR Y . Here 0 ′ denotes the halting problem for Turing machines. See Dobrinen/Simpson [10], Binns/Kjos-Hanssen/Lerman/Solomon [3], Cholak/Greenberg/Miller =-=[7]-=-, Kjos-Hanssen [17], Kjos-Hanssen/Miller/Solomon [18], and our exposition in [42]. The relationship between almost everywhere domination and mass problems was developed in Kjos-Hanssen [17] and Simpso... |

27 |
Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic
- Simpson
- 1999
(Show Context)
Citation Context ...ed in certain formal, deductive systems. The most important formal systems for reverse mathematics are the so-called “Big Five”: RCA0, WKL0, ACA0, ATR0, Π 1 1 -CA0, corresponding to Chapters II–VI of =-=[40, 47]-=-. The standard reference for reverse mathematics is Simpson [40, 47]. See also our recent survey in [50]. The present paper includes a contribution to the reverse mathematics of measure theory. In ord... |

25 |
On a question of Dobrinen and Simpson concerning almost everywhere domination
- Binns, Kjos-Hanssen, et al.
(Show Context)
Citation Context ...y Σ0 3 set includes a Σ0,Y 2 set of the same measure, if and only if 0 ′ ≤LR Y . Here 0 ′ denotes the halting problem for Turing machines. See Dobrinen/Simpson [10], Binns/Kjos-Hanssen/Lerman/Solomon =-=[3]-=-, Cholak/Greenberg/Miller [7], Kjos-Hanssen [17], Kjos-Hanssen/Miller/Solomon [18], and our exposition in [42]. The relationship between almost everywhere domination and mass problems was developed in... |

25 | Low for random reals and positive-measure domination
- Kjos-Hanssen
(Show Context)
Citation Context ...asure, if and only if 0 ′ ≤LR Y . Here 0 ′ denotes the halting problem for Turing machines. See Dobrinen/Simpson [10], Binns/Kjos-Hanssen/Lerman/Solomon [3], Cholak/Greenberg/Miller [7], Kjos-Hanssen =-=[17]-=-, Kjos-Hanssen/Miller/Solomon [18], and our exposition in [42]. The relationship between almost everywhere domination and mass problems was developed in Kjos-Hanssen [17] and Simpson [44]. In particul... |

25 |
On strong and weak reducibility of algorithmic problems
- Muchnik
- 1963
(Show Context)
Citation Context ...ial. We use 1 and 0 to denote the top and bottom degrees in Ew. Remark 1.5. Historically, the study of mass problems and Dw originated in considerations of Kolmogorov [21], Medvedev [24], and Muchnik =-=[25]-=- concerning the so-called “intuitionistic calculus of problems” [21]. In particular Muchnik [25] showed that Dw is a complete Brouwerian lattice. Some recent papers on Ew are [2, 4, 9, 41, 43, 44, 45,... |

23 |
Degrees of difficulty of the mass problems
- Medvedev
- 1955
(Show Context)
Citation Context ...or space is essential. We use 1 and 0 to denote the top and bottom degrees in Ew. Remark 1.5. Historically, the study of mass problems and Dw originated in considerations of Kolmogorov [21], Medvedev =-=[24]-=-, and Muchnik [25] concerning the so-called “intuitionistic calculus of problems” [21]. In particular Muchnik [25] showed that Dw is a complete Brouwerian lattice. Some recent papers on Ew are [2, 4, ... |

22 | An extension of the recursively enumerable Turing degrees
- Simpson
(Show Context)
Citation Context ...itionism, algorithmic randomness, Kolmogorov complexity, resource bounded computational complexity, subrecursive hierarchies, and unsolvable mathematical problems. See in particular our recent papers =-=[9, 41, 43, 44, 45, 46, 48]-=- and our forthcoming treatise [51]. We emphasize that the study of mass problems offers a path along which the study of degrees of unsolvability can return to and reconnect with its roots in the found... |

21 |
Zur Deutung der intuitionistischen Logik,” Mathematische Zeitschrift 35
- Kolmogoroff
- 1932
(Show Context)
Citation Context ...ion to the Cantor space is essential. We use 1 and 0 to denote the top and bottom degrees in Ew. Remark 1.5. Historically, the study of mass problems and Dw originated in considerations of Kolmogorov =-=[21]-=-, Medvedev [24], and Muchnik [25] concerning the so-called “intuitionistic calculus of problems” [21]. In particular Muchnik [25] showed that Dw is a complete Brouwerian lattice. Some recent papers on... |

20 | Almost everywhere domination and superhighness
- Simpson
- 2007
(Show Context)
Citation Context ... ≰ bα. From this we clearly have inf(bα+1,1) ≰ bα and d ≰ inf(bα,1). By the Low Basis Theorem (see for instance [41]) let Z be Martin-Löf random and low, i.e., Z ′ ≤T 0 ′. By Corollary 4.10 (see also =-=[42]-=-) we know that each Y ∈ Bα for α > 0 is high, i.e., 0 ′′ ≤T Y ′. Thus Bα ≰w {Z}. Moreover, in view of Stephan’s Theorem [53] (see also our exposition in [44, Section 6]) we have PA ≰w {Z}. Thus Bα ∪ P... |

17 | Medvedev Degrees of 2-Dimensional Subshifts of Finite Type, to appear, Ergodic Theory and Dynamical Systems
- Simpson
(Show Context)
Citation Context |

10 | Mass problems and almost everywhere domination
- Simpson
(Show Context)
Citation Context |

9 | Some fundamental issues concerning degrees of unsolvability
- Simpson
- 2008
(Show Context)
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7 | Martin-Löf random and PA-complete sets
- Stephan
- 2006
(Show Context)
Citation Context ...be Martin-Löf random and low, i.e., Z ′ ≤T 0 ′. By Corollary 4.10 (see also [42]) we know that each Y ∈ Bα for α > 0 is high, i.e., 0 ′′ ≤T Y ′. Thus Bα ≰w {Z}. Moreover, in view of Stephan’s Theorem =-=[53]-=- (see also our exposition in [44, Section 6]) we have PA ≰w {Z}. Thus Bα ∪ PA ≰w {Z}, and this implies that inf(bα,1) ≰ r. For α > 0 we have seen that inf(bα,1) is ≱ d and ≰ r. From this plus d < r it... |

6 |
FOM: natural r.e. degrees; Pi01 classes. FOM e-mail list [10
- Simpson
- 1999
(Show Context)
Citation Context ... Indeed, the principal axiom of WWKL0 turned out to be equivalent over RCA0 to the statement ∀X ∃Y (Y is Martin-Löf random relative to X). See also our summary in [40, 49, Section X.1]. Later Simpson =-=[38, 39, 41]-=- developed the relationship to mass problems. For instance, the Muchnik degree r = deg w ({X | X is Martin-Löf random}) 4was the first example of a specific, natural degree in Ew other than 0 and 1. ... |

5 | editors. Logic Colloquium ’02 - Chatzidakis, Koepke, et al. |

5 |
Classical Recursion Theory. Number 125
- Odifreddi
- 1989
(Show Context)
Citation Context ...tructural and methodological questions concerning various degree structures. Some classical treatises from this period are Sacks [33], Rogers [32], Shoenfield [36], Lerman [22], Soare [52], Odifreddi =-=[29, 30]-=-. 2The classical theory of degrees of unsolvability was concerned mainly with decision problems and their Turing degrees. A more recent trend has been to focus instead on mass problems and their Much... |

5 |
Classical Recursion Theory, Volume 2. Number 143
- Odifreddi
- 1999
(Show Context)
Citation Context ...tructural and methodological questions concerning various degree structures. Some classical treatises from this period are Sacks [33], Rogers [32], Shoenfield [36], Lerman [22], Soare [52], Odifreddi =-=[29, 30]-=-. 2The classical theory of degrees of unsolvability was concerned mainly with decision problems and their Turing degrees. A more recent trend has been to focus instead on mass problems and their Much... |

5 |
Degrees of Unsolvability. Number 55 in Annals of Mathematics Studies
- Sacks
- 1963
(Show Context)
Citation Context ...954 [20]. Later research during the period 1960–1990 was motivated by structural and methodological questions concerning various degree structures. Some classical treatises from this period are Sacks =-=[33]-=-, Rogers [32], Shoenfield [36], Lerman [22], Soare [52], Odifreddi [29, 30]. 2The classical theory of degrees of unsolvability was concerned mainly with decision problems and their Turing degrees. A ... |

5 |
Measure theory in weak subsystems of second order arithmetic
- Yu
- 1987
(Show Context)
Citation Context ... in this area. The first wave of research in the reverse mathematics of measure theory dealt with additivity properties and was centered around the system WWKL0. This was initiated in the 1980s by Yu =-=[55]-=- and continued in Yu/Simpson [60], Yu [56, 57, 58, 59], and Brown/Giusto/Simpson [5]. Recall the principal axiom of WKL0, which says that any tree T containing bitstrings of length n for each n ∈ N ha... |

5 |
Lebesgue convergence theorems and reverse mathematics
- Yu
- 1994
(Show Context)
Citation Context ...in the reverse mathematics of measure theory dealt with additivity properties and was centered around the system WWKL0. This was initiated in the 1980s by Yu [55] and continued in Yu/Simpson [60], Yu =-=[56, 57, 58, 59]-=-, and Brown/Giusto/Simpson [5]. Recall the principal axiom of WKL0, which says that any tree T containing bitstrings of length n for each n ∈ N has an infinite path. The principal axiom of WWKL0 is we... |

5 |
Measure theory and weak König’s lemma
- Yu, Simpson
- 1990
(Show Context)
Citation Context ...research in the reverse mathematics of measure theory dealt with additivity properties and was centered around the system WWKL0. This was initiated in the 1980s by Yu [55] and continued in Yu/Simpson =-=[60]-=-, Yu [56, 57, 58, 59], and Brown/Giusto/Simpson [5]. Recall the principal axiom of WKL0, which says that any tree T containing bitstrings of length n for each n ∈ N has an infinite path. The principal... |

4 |
Set existence. Bulletin de l’Academie Polonaise des
- Gandy, Kreisel, et al.
- 1960
(Show Context)
Citation Context ...uired. For M3 we need another lemma: Lemma 7.6. Given Y and Ai ≰T Y for all i, we can find Z such that Z is PA-complete over Y and Ai ≰T Y ⊕ Z for all i. Proof. This is the Gandy/Kreisel/Tait Theorem =-=[14]-=-. See also our exposition in [40, 49, Theorem VIII.2.2.4]. To build M3 start with ∀i (Ai ≰T 0) and apply Theorem 5.8 and Lemma 7.6 repeatedly for n = 0, 1, 2, . . . to obtain Yn ∈ M and Zn ∈ M such th... |

4 |
FOM: priority arguments; Kleene-r.e. degrees; Pi01 classes. FOM e-mail list [10
- Simpson
- 1999
(Show Context)
Citation Context ... Indeed, the principal axiom of WWKL0 turned out to be equivalent over RCA0 to the statement ∀X ∃Y (Y is Martin-Löf random relative to X). See also our summary in [40, 49, Section X.1]. Later Simpson =-=[38, 39, 41]-=- developed the relationship to mass problems. For instance, the Muchnik degree r = deg w ({X | X is Martin-Löf random}) 4was the first example of a specific, natural degree in Ew other than 0 and 1. ... |

4 | The Gödel hierarchy and reverse mathematics
- Simpson
(Show Context)
Citation Context ...o-called “Big Five”: RCA0, WKL0, ACA0, ATR0, Π 1 1 -CA0, corresponding to Chapters II–VI of [40, 49]. The standard reference for reverse mathematics is Simpson [40, 49]. See also our recent survey in =-=[47]-=-. The present paper includes a contribution to the reverse mathematics of measure theory. In order to place this contribution in context, we now briefly outline the previous research in this area. The... |

4 |
Riesz representation theorem, Borel measures and subsystems of second-order arithmetic
- Yu
- 1993
(Show Context)
Citation Context ...in the reverse mathematics of measure theory dealt with additivity properties and was centered around the system WWKL0. This was initiated in the 1980s by Yu [55] and continued in Yu/Simpson [60], Yu =-=[56, 57, 58, 59]-=-, and Brown/Giusto/Simpson [5]. Recall the principal axiom of WKL0, which says that any tree T containing bitstrings of length n for each n ∈ N has an infinite path. The principal axiom of WWKL0 is we... |

3 | Logic and Computation. Contemporary Mathematics - Sieg, editor - 1990 |

3 | Mass problems and intuitionism
- Simpson
- 2007
(Show Context)
Citation Context |

3 |
Radon-Nikodym theorem is equivalent to arithmetical comprehension
- Yu
- 1990
(Show Context)
Citation Context ...in the reverse mathematics of measure theory dealt with additivity properties and was centered around the system WWKL0. This was initiated in the 1980s by Yu [55] and continued in Yu/Simpson [60], Yu =-=[56, 57, 58, 59]-=-, and Brown/Giusto/Simpson [5]. Recall the principal axiom of WKL0, which says that any tree T containing bitstrings of length n for each n ∈ N has an infinite path. The principal axiom of WWKL0 is we... |

3 | editors. Kurt Gödel: Essays for his Centennial. Number 33 - Feferman, Parsons, et al. - 2010 |

2 |
Degrees of Unsolvability, volume 2 of North-Holland Mathematics Studies
- Shoenfield
- 1971
(Show Context)
Citation Context ...g the period 1960–1990 was motivated by structural and methodological questions concerning various degree structures. Some classical treatises from this period are Sacks [33], Rogers [32], Shoenfield =-=[36]-=-, Lerman [22], Soare [52], Odifreddi [29, 30]. 2The classical theory of degrees of unsolvability was concerned mainly with decision problems and their Turing degrees. A more recent trend has been to ... |