A 1-generic degree which bounds a minimal degree

1990 ◽  
Vol 55 (2) ◽  
pp. 733-743 ◽  
Author(s):  
Masahiro Kumabe
Keyword(s):  
Affirmative Answer ◽  
Minimal Degree ◽  
Turing Degree ◽  
Natural Numbers ◽  
Degree Bounds

Let ω be the set of natural numbers, i.e. {0,1,2,…}. A set A (≤ω) is called n-generic if it is Cohen-generic for n-quantifier arithmetic. As characterized by Jockusch [4], this is equivalent to saying that for every set of strings S, there is a σ < A such that σ ∈ S or ∀ν ≥ σ(ν ∉ S). When we say degree, we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. A nonrecursive degree a is called minimal if there is no nonrecursive degree b with b < a. Jockusch [4] exhibited various properties of generic degrees, and he showed that any 2-generic degree bounds no minimal degree. Chong and Jockusch [1] showed that any 1-generic degree below 0′ bounds no minimal degree. Haught [3] refuted one of the conjectures in [1] and showed that if a is a 1-generic degree and 0 < b < a < 0′ then b is also 1-generic. We show here that there is a 1-generic degree which bounds a minimal degree. This gives an affirmative answer to questions in [1] and [4], As any 1-generic degree below 0′ bounds no minimal degree, we see that our 1-generic degree which bounds a minimal degree is not below 0′, but can be constructed recursively in 0″. Furthermore we see that the initial segments below 1-generic degrees are not order isomorphic.

Relative recursive enumerability of generic degrees

1991 ◽  
Vol 56 (3) ◽  
pp. 1075-1084 ◽  
Author(s):  
Masahiro Kumabe
Keyword(s):  
Characteristic Function ◽  
Initial Segment ◽  
Minimal Degree ◽  
Turing Degree ◽  
Natural Numbers ◽  
Recursive Enumerability ◽  
Degree Bounds ◽  
Generic Set ◽  
The Given ◽  
Dense Set

Let ω be the set of natural numbers, i.e. {0, 1, 2, 3, …}. A string is a mapping from an initial segment of ω into {0, 1}. We identify a set A ≤ ω with its characteristic function. A set A ≤ ω is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every set of strings S, there is a σ < A such that σ ∈ S or (∀ν ≥ σ)(ν ∉ S). By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, D(≤ a) denotes the set of degrees recursive in a.The relation between generic degrees and minimal degrees has been widely studied. Spector [9] proved the existence of minimal degrees. Shoenfield [8] simplified the proof by using trees. In the construction of a minimal degree, given σ we extend σ to ν so that ν is in the (splitting or nonsplitting) subtree of a given tree. But in the construction of a generic set, given σ we extend σ to ν to meet the given dense set. So these two constructions are quite different. Jockusch [5] showed that any 2-generic degree bounds no minimal degree. Chong and Jockusch [3] showed that any 1-generic degree below 0′ bounds no minimal degree.

A note on degrees of subsets1

1969 ◽  
Vol 34 (2) ◽  
pp. 256-256 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Minimal Degree ◽  
Turing Degree ◽  
Lower Degree ◽  
Natural Numbers ◽  
Infinite Set

In [2] we constructed an infinite set of natural numbers containing no subset of higher (Turing) degree. Since it is well known that there are nonrecursive sets (e.g. sets of minimal degree) containing no nonrecursive subset of lower degree, it is natural to suppose that these arguments may be combined, but this is false. We prove that every infinite set must contain a nonrecursive subset of either higher or lower degree.

scholarly journals FINDING THE LIMIT OF INCOMPLETENESS I

2020 ◽  
Vol 26 (3-4) ◽  
pp. 268-286
Author(s):  
YONG CHENG
Keyword(s):  
Model Theory ◽  
Recursion Theory ◽  
Minimal Degree ◽  
Incompleteness Theorem ◽  
Turing Degree ◽  
Turing Reducibility ◽  
Inseparable Pair

AbstractIn this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

Degrees of classes of RE sets

1976 ◽  
Vol 41 (3) ◽  
pp. 695-696 ◽  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Order Theory ◽  
Turing Degree ◽  
Natural Numbers ◽  
First Order ◽  
First Order Theory ◽  
Simple Sets ◽  
Natural Classes

In [3], Martin computed the degrees of certain classes of RE sets. To state the results succinctly, some notation is useful.If A is a set (of natural numbers), dg(A) is the (Turing) degree of A. If A is a class of sets, dg(A) = {dg(A): A ∈ A). Let M be the class of maximal sets, HHS the class of hyperhypersimple sets, and DS the class of dense simple sets. Martin showed that dg(M), dg(HHS), and dg(DS) are all equal to the set H of RE degrees a such that a′ = 0″.Let M* be the class of coinfinite RE sets having no superset in M; and define HHS* and DS* similarly. Martin showed that dg(DS*) = H. In [2], Lachlan showed (among other things) that dg(M*)⊆K, where K is the set of RE degrees a such that a″ > 0″. We will show that K ⊆ dg (HHS*). Since maximal sets are hyperhypersimple, this gives dg(M*) = dg (HHS*) = K.These results suggest a problem. In each case in which dg(A) has been calculated, the set of nonzero degrees in dg(A) is either H or K or the empty set or the set of all nonzero RE degrees. Is this always the case for natural classes A? Natural here might mean that A is invariant under all automorphisms of the lattice of RE sets; or that A is definable in the first-order theory of that lattice; or anything else which seems reasonable.

Minimal complements for degrees below 0′

2004 ◽  
Vol 69 (4) ◽  
pp. 937-966 ◽  
Author(s):  
Andrew Lewis
Keyword(s):  
Minimal Degree ◽  
Turing Degree

Abstract.It is shown that for every (Turing) degree 0 < a < 0′ there is a minimal degree m < 0′ such that a ∨ m = 0′ (and therefore a ∧ m = 0).

A 1-generic degree with a strong minimal cover

2000 ◽  
Vol 65 (3) ◽  
pp. 1395-1442 ◽  
Author(s):  
Masahiro Kumabe
Keyword(s):  
Turing Degree ◽  
Natural Numbers ◽  
Minimal Cover ◽  
Recursively Enumerable

We consider a set generic over the arithmetic sets. A subset A of the natural numbers is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every -set of strings S, there is a string σ ⊂ A such that σ ∈ S or no extension of σ is in S. By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, let D(≤ a) denote the set of degrees which are recursive in a.We say a is a strong minimal cover of g if every degree strictly below a is less than or equal to g. In this paper we show that there are a degree a and a 1-generic degree g < a such that a is a strong minimal cover of g. This easily implies that there is a 1-generic degree without the cupping property. Jockusch [7] showed that every 2-generic degree has the cupping property. Slaman and Steel [17] and independently Cooper [3] showed that there are recursively enumerable degrees a and b < a such that no degree c < a joins b above a. Take a 1-generic degree g below b. Then g does not have the cupping property.

scholarly journals Degree bounds for modular covariants

2020 ◽  
Vol 32 (4) ◽  
pp. 905-910
Author(s):  
Jonathan Elmer ◽  
Müfit Sezer
Keyword(s):  
Cyclic Group ◽  
Prime Order ◽  
Minimal Degree ◽  
Polynomial Maps ◽  
Degree Bounds

AbstractLet {V,W} be representations of a cyclic group G of prime order p over a field {\Bbbk} of characteristic p. The module of covariants {\Bbbk[V,W]^{G}} is the set of G-equivariant polynomial maps {V\rightarrow W}, and is a module over {\Bbbk[V]^{G}}. We give a formula for the Noether bound {\beta(\Bbbk[V,W]^{G},\Bbbk[V]^{G})}, i.e. the minimal degree d such that {\Bbbk[V,W]^{G}} is generated over {\Bbbk[V]^{G}} by elements of degree at most d.

Nonisomorphism of lattices of recursively enumerable sets

1993 ◽  
Vol 58 (4) ◽  
pp. 1177-1188 ◽  
Author(s):  
John Todd Hammond
Keyword(s):  
Boolean Algebra ◽  
Isomorphism Class ◽  
Boolean Algebras ◽  
Turing Degree ◽  
Symmetric Difference ◽  
Natural Numbers ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Definition Of

Let ω be the set of natural numbers, let be the lattice of recursively enumerable subsets of ω, and let A be the lattice of subsets of ω which are recursively enumerable in A. If U, V ⊆ ω, put U =* V if the symmetric difference of U and V is finite.A natural and interesting question is then to discover what the relation is between the Turing degree of A and the isomorphism class of A. The first result of this form was by Lachlan, who proved [6] that there is a set A ⊆ ω such that A ≇ . He did this by finding a set A ⊆ ω and a set C ϵ A such that the structure ({W ϵ A∣W ⊇ C},∪,∩)/=* is a Boolean algebra and is not isomorphic to the structure ({W ϵ ∣W ⊇ D},∪,∩)/=* for any D ϵ . There is a nonrecursive ordinal which is recursive in the set A which he constructs, so his set A is not (see, for example, Shoenfield [11] for a definition of what it means for a set A ⊆ ω to be ). Feiner then improved this result substantially by proving [1] that for any B ⊆ ω, B′ ≇ B, where B′ is the Turing jump of B. To do this, he showed that for each X ⊆= ω there is a Boolean algebra which is but not and then applied a theorem of Lachlan [6] (definitions of and Boolean algebras will be given in §2). Feiner's result is of particular interest for the case B = ⊘, for it shows that the set A of Lachlan can actually be chosen to be arithmetical (in fact, ⊘′), answering a question that Lachlan posed in his paper. Little else has been known.

Sets with no subset of higher degree

1969 ◽  
Vol 34 (1) ◽  
pp. 53-56 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Turing Degree ◽  
Natural Numbers ◽  
Original Proof ◽  
Infinite Set

The problem of finding an infinite set of natural numbers which contains no subsets of higher (Turing) degree was first posed by W. Miller [3] and was brought to our attention by C. G. Jockusch, Jr., who proved that such a set, if it existed, could not be hyperarithmetic.2 In this paper we construct an infinite set which is not recursive in any of its coinfinite subsets, and thus contains no subset of higher degree. Our original proof made use of the result (attributed to Ehrenfeucht) that every subset of 2ω which is open (in the standard topology) is “Ramsey”.

Relative enumerability and 1-genericity

2011 ◽  
Vol 76 (3) ◽  
pp. 897-913 ◽  
Author(s):  
Wei Wang
Keyword(s):  
Turing Degree ◽  
Natural Numbers ◽  
Relative Enumerability ◽  
Computably Enumerable

AbstractA set of natural numbers B is computably enumerable in and strictly above (or c.e.a. for short) another set C if C <TB and B is computably enumerable in C. A Turing degree b is c.e.a. c if b and c respectively contain B and C as above. In this paper, it is shown that if b is c.e.a. c then b is c.e.a. some 1-generic g.

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