## Some fundamental issues concerning degrees of unsolvability (2007)

Venue: | In [6], 2005. Preprint |

Citations: | 9 - 8 self |

### BibTeX

@INPROCEEDINGS{Simpson07somefundamental,

author = {Stephen G. Simpson},

title = {Some fundamental issues concerning degrees of unsolvability},

booktitle = {In [6], 2005. Preprint},

year = {2007},

pages = {53--462}

}

### OpenURL

### Abstract

Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a “smallness property ” of an infinite, co-recursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is> 0 and < 0 ′. In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.

### Citations

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Theory of recursive functions and effective computability
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(Show Context)
Citation Context ...he recur1ssively enumerable Turing degrees. Both of these semilattices have been principal objects of study in recursion theory for many decades. See for instance the monographs of Sacks [39], Rogers =-=[38]-=-, Soare [45], and Odifreddi [35, 36]. Two fundamental, classical, unresolved issues concerning RT are: Issue 1. To find a specific, natural, recursively enumerable Turing degree a ∈ RT which is > 0 an... |

473 | Recursively Enumerable Sets and Degrees - Soare - 1987 |

332 |
The definition of random sequences
- Martin-Löf
- 1966
(Show Context)
Citation Context ...arisen in the reverse mathematics of measure theory and of the Tietze Extension Theorem, respectively [10, 19]. See also [43, Chapter X]. Remark. We hereby assign the names “Carl”, “Klaus”, and “Per” =-=[22, 3, 30]-=- to the respective weak degrees d, dREC, and r1 in Pw. The Embedding Lemma and some of its consequences Several of the results stated above are consequences of the following lemma, due to Simpson [42,... |

193 | Subsystems of Second Order Arithmetic - Simpson - 2009 |

126 |
Π 0 1 classes and degrees of theories
- Soare
- 1979
(Show Context)
Citation Context ...y bounded, Π 0 1 subsets of ωω . There is an obvious partial ordering of Pw induced by weak reducibility. Thus deg w(P) ≤ deg w(Q) if and only if P ≤w Q. Remark. Many authors including Jockusch/Soare =-=[23]-=- and Groszek/Slaman [20] have studied the Turing degrees of elements of Π 0 1 subsets of ωω which are nonempty and recursively bounded. This earlier research is part of the inspiration for our current... |

117 |
Recursively enumerable sets of positive integers and their decision problems
- Post
- 1944
(Show Context)
Citation Context ...at the Turing degree deg T (A) = a ∈ RT is > 0 and < 0 ′ . These unresolved issues go back to Post’s classical 1944 paper, Recursively enumerable sets of positive integers and their decision problems =-=[37]-=-. My recent interest in Issue 1 began in 1999 at a conference in Boulder, Colorado [11]. There I heard a talk by Shmuel Weinberger, a prominent topologist and geometer. At the time Weinberger was tryi... |

57 |
Zum Hilbertschen Aufbau der reellen Zahlen
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- 1928
(Show Context)
Citation Context ...· · > dα > dα+1 > · · · > dε0 > dREC in Pw. Moreover, if α is a limit ordinal, then by [44, Remark 10.12] we have dα = infβ<α dα. Remark. We hereby assign the names “Wilhelm”, “László”, and “Stanley” =-=[1, 25, 47]-=- to the respective weak degrees d0 = dPR, dER, and dε0 in Pw. In addition, let d 2 be the weak degree of the set of f ⊕ g such that f is diagonally nonrecursive, and g is diagonally nonrecursive relat... |

46 | Degrees of Random Sets
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- 1991
(Show Context)
Citation Context ...rsively enumerable Turing degrees, except 0 ′ and 0. Some specific, natural degrees in Pw Conside the following specific, natural, weak degrees. Let rn be the weak degree of the set of n-random reals =-=[26]-=-. Let d be the weak degree of the set of diagonally nonrecursive functions [22]. Let dREC be the weak degree of the set of diagonally nonrecursive functions which are recursively bounded. Not all of t... |

36 | problems and randomness
- Simpson
(Show Context)
Citation Context ...ch that a < c < b. 4. There are some degrees in Pw with interesting lattice-theoretic properties, such as being meet-reducible or not, and joining to 0 ′ or not. See Theorem 3 below. See also Simpson =-=[42, 44]-=-. Note that these structural properties of Pw are proved by means of priority arguments, just as for RT. On the other hand, there are some structural differences between Pw and RT. For example: 5. Wit... |

34 | A splitting theorem for the Medvedev and Muchnik lattices
- Binns
(Show Context)
Citation Context ... 9]. 2. The Pw analog of the Sacks Splittting Theorem holds. In other words, for all a,c > 0 in Pw we can find b1,b2 ∈ Pw such that a = sup(b1,b2) and b1 �≥ c and b2 �≥ c. This result is due to Binns =-=[4, 5]-=-. 3. We conjecture that the Pw analog of the Sacks Density Theorem holds. This would mean that for all a,b ∈ Pw with a < b there exists c ∈ Pw such that a < c < b. 4. There are some degrees in Pw with... |

33 | Almost everywhere domination
- Dobrinen, Simpson
(Show Context)
Citation Context ...ned out to be incomparable with all of the recursively enumerable Turing degrees, except 0 ′ and 0. We now present an example which behaves differently in this respect. Starting with Dobrinen/Simpson =-=[15]-=- and continuing with Cholak/Greenberg/Miller [13], Binns et al [7], and Kjos-Hanssen [27], there has been a recent upsurge of interest in domination properties related to the reverse mathematics of me... |

30 |
Array nonrecursive sets and multiple permitting arguments
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- 1990
(Show Context)
Citation Context ...ans of priority arguments. Much is known about them. For example, any two such sets are automorphic in the lattice of Π0 1 subsets of 2ω under inclusion. See Martin/Pour-El [29], Downey/Jockusch/Stob =-=[16, 17]-=-, and Cholak et al [12]. Theorem 6. Let p = deg w(P) where P ⊆ 2 ω is Π 0 1, nonempty, thin, and perfect. Then p is incomparable with r1. Hence 0 < p < 0 ′ . Proof. See Simpson [44]. One may also cons... |

25 |
On a question of Dobrinen and Simpson concerning almost everywhere domination
- Binns, Kjos-Hanssen, et al.
- 1928
(Show Context)
Citation Context ...uring degrees, except 0 ′ and 0. We now present an example which behaves differently in this respect. Starting with Dobrinen/Simpson [15] and continuing with Cholak/Greenberg/Miller [13], Binns et al =-=[7]-=-, and Kjos-Hanssen [27], there has been a recent upsurge of interest in domination properties related to the reverse mathematics of measure theory. We consider one such property. Definition 5. A ∈ 2ω ... |

25 | Uniform almost everywhere domination
- Cholak, Greenberg, et al.
- 2006
(Show Context)
Citation Context ...ively enumerable Turing degrees, except 0 ′ and 0. We now present an example which behaves differently in this respect. Starting with Dobrinen/Simpson [15] and continuing with Cholak/Greenberg/Miller =-=[13]-=-, Binns et al [7], and Kjos-Hanssen [27], there has been a recent upsurge of interest in domination properties related to the reverse mathematics of measure theory. We consider one such property. Defi... |

24 | Array nonrecursive degrees and genericity
- Downey, Jockusch, et al.
- 1996
(Show Context)
Citation Context ...ans of priority arguments. Much is known about them. For example, any two such sets are automorphic in the lattice of Π0 1 subsets of 2ω under inclusion. See Martin/Pour-El [29], Downey/Jockusch/Stob =-=[16, 17]-=-, and Cholak et al [12]. Theorem 6. Let p = deg w(P) where P ⊆ 2 ω is Π 0 1, nonempty, thin, and perfect. Then p is incomparable with r1. Hence 0 < p < 0 ′ . Proof. See Simpson [44]. One may also cons... |

24 |
On strong and weak reducibilities of algorithmic problems
- Muchnik
- 1963
(Show Context)
Citation Context ...ucibility = strong reducibility truth-table reducibility . 3sIn this paper we shall deal only with weak reducibility. As a historical note, we mention that weak reducibility goes back to Muchnik 1963 =-=[32]-=-, while strong reducibility goes back to Medvedev 1955 [31]. Actually, as mentioned by Terwijn [46], both of these notions ultimately derive from ideas concerning the Brouwer/Heyting/Kolmogorov interp... |

23 |
Degrees of functions with no fixed points
- Jockusch
- 1989
(Show Context)
Citation Context ...rees in Pw Conside the following specific, natural, weak degrees. Let rn be the weak degree of the set of n-random reals [26]. Let d be the weak degree of the set of diagonally nonrecursive functions =-=[22]-=-. Let dREC be the weak degree of the set of diagonally nonrecursive functions which are recursively bounded. Not all of these weak degrees belong to Pw. However, we have the following theorem. Theorem... |

22 | Embeddings into the Medvedev and Muchnik lattices of Π0 1 classes, Archive for
- Binns, Simpson
(Show Context)
Citation Context ...res are as follows. 1. Pw is a countable distributive lattice. Moreover, every countable distributive lattice is lattice embeddable in every initial segment of Pw. This result is due to Binns/Simpson =-=[4, 9]-=-. 2. The Pw analog of the Sacks Splittting Theorem holds. In other words, for all a,c > 0 in Pw we can find b1,b2 ∈ Pw such that a = sup(b1,b2) and b1 �≥ c and b2 �≥ c. This result is due to Binns [4,... |

22 |
Degrees of Difficulty of Mass Problems
- Medvedev
- 1955
(Show Context)
Citation Context ...3sIn this paper we shall deal only with weak reducibility. As a historical note, we mention that weak reducibility goes back to Muchnik 1963 [32], while strong reducibility goes back to Medvedev 1955 =-=[31]-=-. Actually, as mentioned by Terwijn [46], both of these notions ultimately derive from ideas concerning the Brouwer/Heyting/Kolmogorov interpretation of intuitionistic propositional calculus. The latt... |

22 | An extension of the recursively enumerable Turing degrees
- Simpson
(Show Context)
Citation Context ...lattice of Turing degrees of recursively enumerable subsets of ω, and Pw = the lattice of weak degrees of nonempty Π0 1 subsets of 2ω . The following embedding theorem was obtained by Simpson in 2002 =-=[42]-=-. 5 Y QsTheorem 1. There is a specific, natural embedding The embedding φ is given by φ : RT ֒→ Pw . φ : deg T(A) ↦→ deg w(PA ∪ {A}) . Here φ is one-to-one and preserves the partial ordering ≤, the le... |

18 | Almost everywhere domination and superhighness
- Simpson
(Show Context)
Citation Context ...d Solomon [8] have shown that A ∈ 2 ω is positive-measure dominating if and only if A is almost everywhere dominating, if and only if A is uniformly almost everywhere dominating. In addition, Simpson =-=[41]-=- has shown that any such A is superhigh, i.e., A ′ ≥tt 0 ′′ , i.e., 0 ′′ is truth-table reducible to A ′ . Some further specific, natural degrees in Pw We now mention some further examples of specific... |

16 | Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees
- Cholak, Coles, et al.
(Show Context)
Citation Context ...Much is known about them. For example, any two such sets are automorphic in the lattice of Π0 1 subsets of 2ω under inclusion. See Martin/Pour-El [29], Downey/Jockusch/Stob [16, 17], and Cholak et al =-=[12]-=-. Theorem 6. Let p = deg w(P) where P ⊆ 2 ω is Π 0 1, nonempty, thin, and perfect. Then p is incomparable with r1. Hence 0 < p < 0 ′ . Proof. See Simpson [44]. One may also consider other smallness pr... |

16 |
Axiomatizable theories with few axiomatizable extensions
- Martin, Pour-El
- 1970
(Show Context)
Citation Context ...have been constructed by means of priority arguments. Much is known about them. For example, any two such sets are automorphic in the lattice of Π0 1 subsets of 2ω under inclusion. See Martin/Pour-El =-=[29]-=-, Downey/Jockusch/Stob [16, 17], and Cholak et al [12]. Theorem 6. Let p = deg w(P) where P ⊆ 2 ω is Π 0 1, nonempty, thin, and perfect. Then p is incomparable with r1. Hence 0 < p < 0 ′ . Proof. See ... |

15 | Lowness notions, measure and domination
- Kjos-Hanssen, Miller, et al.
- 2012
(Show Context)
Citation Context ...< 0 ′ in Pw. See Figure 4. Remark. We hereby assign the name “Bjørn” [27] to the specific, natural degree inf(m,0 ′ ) in Pw. Note added June 30, 2006: Recently Binns, Kjos-Hanssen, Miller and Solomon =-=[8]-=- have shown that A ∈ 2 ω is positive-measure dominating if and only if A is almost everywhere dominating, if and only if A is uniformly almost everywhere dominating. In addition, Simpson [41] has show... |

13 | Located sets and reverse mathematics
- Giusto, Simpson
(Show Context)
Citation Context ...le and does not join to 0 ′ . Proof. See Simpson [42, 44]. Remark. The weak degrees r1 and d have arisen in the reverse mathematics of measure theory and of the Tietze Extension Theorem, respectively =-=[10, 19]-=-. See also [43, Chapter X]. Remark. We hereby assign the names “Carl”, “Klaus”, and “Per” [22, 3, 30] to the respective weak degrees d, dREC, and r1 in Pw. The Embedding Lemma and some of its conseque... |

13 |
Correction to “a cohesive set which is not high
- Stephan
- 1997
(Show Context)
Citation Context ...er, if Q is specific and natural, then so is s, hence so is inf(s,0 ′ ) ∈ Pw. It should be interesting to explore the relationships among these degrees and others in Pw. The ideas of Jockusch/Stephan =-=[24]-=- and Kjos-Hanssen [7, 27] concerning cohesiveness and superhighness may be relevant. Response to Issue 2 Issue 2 was the problem of finding a “smallness property” of infinite Π 0 1 (i.e., co-recursive... |

12 |
A classification of the ordinal recursive functions. Archiv für
- Wainer
- 1970
(Show Context)
Citation Context ...able functions. 5. EXPTIME = the class of exponential-time computable functions, etc. 6. Cα = the class of recursive functions at levels < ω·(1+α) of the transfinite Ackermann hierarchy due to Wainer =-=[47]-=-. Here α is any ordinal number ≤ ε0. Thus C0 = PR, C1 = the class of functions which are primitive recursive in the Ackermann function, etc. For each of these classes C, we have a specific, natural de... |

9 |
The Medvedev and Muchnik Lattices of Π 0 1 Classes
- Binns
- 2003
(Show Context)
Citation Context ...res are as follows. 1. Pw is a countable distributive lattice. Moreover, every countable distributive lattice is lattice embeddable in every initial segment of Pw. This result is due to Binns/Simpson =-=[4, 9]-=-. 2. The Pw analog of the Sacks Splittting Theorem holds. In other words, for all a,c > 0 in Pw we can find b1,b2 ∈ Pw such that a = sup(b1,b2) and b1 �≥ c and b2 �≥ c. This result is due to Binns [4,... |

9 |
Einstein structures: existence versus uniqueness
- Nabutovsky
- 1995
(Show Context)
Citation Context ...bout the recursively enumerable Turing degrees, RT, with an eye to applying them in the study of moduli spaces in differential geometry [48], using recursion-theoretic methods pioneered by Nabutovsky =-=[33, 34]-=-. Weinberger was visibly frustrated by the fact that RT does not appear to contain any specific, natural examples of recursively enumerable Turing degrees, beyond the two standard examples due to Turi... |

8 |
A fixed point free minimal degree
- Kumabe, Lewis
(Show Context)
Citation Context ..., natural degrees in Pw are incomparable with all of the recursively enumerable Turing degrees, except 0 ′ and 0. Proof. See Simpson [42, 44]. The strict inequalities d < dREC < r1 follow from Kumabe =-=[28]-=- and Ambos-Spies et al [3]. 8sWe also have: Theorem 3. 1. We may characterize r1 as the maximum weak degree of a Π 0 1 2 ω which is of positive measure. subset of 2. We may characterize inf(r2,0 ′ ) a... |

8 |
The Medvedev lattice of computably closed sets
- Terwijn
(Show Context)
Citation Context ...weak reducibility. As a historical note, we mention that weak reducibility goes back to Muchnik 1963 [32], while strong reducibility goes back to Medvedev 1955 [31]. Actually, as mentioned by Terwijn =-=[46]-=-, both of these notions ultimately derive from ideas concerning the Brouwer/Heyting/Kolmogorov interpretation of intuitionistic propositional calculus. The lattice Pw (rigorous definition) Recall that... |

7 |
Moduli: The Large-Scale Fractal Geometry of Riemannian Moduli Space
- Computers
- 2005
(Show Context)
Citation Context ...geometer. At the time Weinberger was trying to learn something about the recursively enumerable Turing degrees, RT, with an eye to applying them in the study of moduli spaces in differential geometry =-=[48]-=-, using recursion-theoretic methods pioneered by Nabutovsky [33, 34]. Weinberger was visibly frustrated by the fact that RT does not appear to contain any specific, natural examples of recursively enu... |

6 |
classes. Archive for Mathematical Logic
- Binns
- 2006
(Show Context)
Citation Context ...ence 0 < p < 0 ′ . Proof. See Simpson [44]. One may also consider other smallness properties. As above, let P be a nonempty Π 0 1 subset of 2 ω . The following definition and theorem are due to Binns =-=[6]-=-. Definition 7. P is small if there is no recursive function f such that for all n there exist n members of P which differ at level f(n) in the binary tree {0, 1} <ω . For example, let A ⊆ ω be hypers... |

5 |
Classical Recursion Theory. Number 125
- Odifreddi
- 1989
(Show Context)
Citation Context ...ng degrees. Both of these semilattices have been principal objects of study in recursion theory for many decades. See for instance the monographs of Sacks [39], Rogers [38], Soare [45], and Odifreddi =-=[35, 36]-=-. Two fundamental, classical, unresolved issues concerning RT are: Issue 1. To find a specific, natural, recursively enumerable Turing degree a ∈ RT which is > 0 and < 0 ′ . Issue 2. To find a “smalln... |

5 |
Classical Recursion Theory, Volume 2. Number 143
- Odifreddi
- 1999
(Show Context)
Citation Context ...ng degrees. Both of these semilattices have been principal objects of study in recursion theory for many decades. See for instance the monographs of Sacks [39], Rogers [38], Soare [45], and Odifreddi =-=[35, 36]-=-. Two fundamental, classical, unresolved issues concerning RT are: Issue 1. To find a specific, natural, recursively enumerable Turing degree a ∈ RT which is > 0 and < 0 ′ . Issue 2. To find a “smalln... |

5 |
Degrees of Unsolvability. Number 55 in Annals of Mathematics Studies
- Sacks
- 1963
(Show Context)
Citation Context ...nsisting of the recur1ssively enumerable Turing degrees. Both of these semilattices have been principal objects of study in recursion theory for many decades. See for instance the monographs of Sacks =-=[39]-=-, Rogers [38], Soare [45], and Odifreddi [35, 36]. Two fundamental, classical, unresolved issues concerning RT are: Issue 1. To find a specific, natural, recursively enumerable Turing degree a ∈ RT wh... |

5 |
On the degrees of complete extensions of arithmetic (abstract
- Scott, Tennenbaum
- 1960
(Show Context)
Citation Context ...f Dw. 2. The bottom element of Pw is 0, the same as the bottom element of Dw. 3. The top element of Pw is the weak degree of PA = {completions of Peano Arithmetic}. This goes back to Scott/Tennenbaum =-=[40]-=-. See also Jockusch/Soare [23]. Remark. RT is usually regarded as the smallest or simplest natural subsemilattice of DT. Similarly, Pw may be regarded as the smallest or simplest natural sublattice of... |

4 |
Π 0 1 classes and minimal degrees
- Groszek, Slaman
- 1997
(Show Context)
Citation Context ... of ωω . There is an obvious partial ordering of Pw induced by weak reducibility. Thus deg w(P) ≤ deg w(Q) if and only if P ≤w Q. Remark. Many authors including Jockusch/Soare [23] and Groszek/Slaman =-=[20]-=- have studied the Turing degrees of elements of Π 0 1 subsets of ωω which are nonempty and recursively bounded. This earlier research is part of the inspiration for our current study of the weak degre... |

3 |
Fundamental group and contractible closed geodesics
- Nabutovsky
- 1996
(Show Context)
Citation Context ...bout the recursively enumerable Turing degrees, RT, with an eye to applying them in the study of moduli spaces in differential geometry [48], using recursion-theoretic methods pioneered by Nabutovsky =-=[33, 34]-=-. Weinberger was visibly frustrated by the fact that RT does not appear to contain any specific, natural examples of recursively enumerable Turing degrees, beyond the two standard examples due to Turi... |

2 |
now Tamara Lakins). Effective Versions of Ramsey’s Theorem
- Hummel
- 1993
(Show Context)
Citation Context .... Let d ∗ be the weak degree of the set of functions which are diagonally nonrecursive relative to the Halting Problem. This set of functions has arisen in the reverse mathematics of Ramsey’s Theorem =-=[21]-=-. The Embedding Lemma tells us that inf(d ∗ ,0 ′ ) ∈ Pw. 2. Let d∗ REC = the weak degree of the set of functions which are (a) diagonally nonrecursive relative to the Halting Problem, and (b) recursiv... |

2 |
Egyszerű példa eldönthetetlen aritmetikai problémára. Matematikai és Fizikai Lapok
- László
- 1943
(Show Context)
Citation Context ...· · > dα > dα+1 > · · · > dε0 > dREC in Pw. Moreover, if α is a limit ordinal, then by [44, Remark 10.12] we have dα = infβ<α dα. Remark. We hereby assign the names “Wilhelm”, “László”, and “Stanley” =-=[1, 25, 47]-=- to the respective weak degrees d0 = dPR, dER, and dε0 in Pw. In addition, let d 2 be the weak degree of the set of f ⊕ g such that f is diagonally nonrecursive, and g is diagonally nonrecursive relat... |

1 |
Low for random reals and positive measure domination
- Kjos-Hanssen
(Show Context)
Citation Context ...0 ′ and 0. We now present an example which behaves differently in this respect. Starting with Dobrinen/Simpson [15] and continuing with Cholak/Greenberg/Miller [13], Binns et al [7], and Kjos-Hanssen =-=[27]-=-, there has been a recent upsurge of interest in domination properties related to the reverse mathematics of measure theory. We consider one such property. Definition 5. A ∈ 2ω is said to be positive-... |