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”.

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.

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.

Proceed with care, because some infinities are larger than others

2020 ◽  
pp. 281-292
Author(s):  
Susan D'Agostino
Keyword(s):  
Number Line ◽  
Mathematics Students ◽  
Natural Numbers ◽  
Real Numbers ◽  
Countable Infinity ◽  
One To One ◽  
Infinite Set

“Proceed with care, because some infinities are larger than others” explains in detail why the infinite set of real numbers—all of the numbers on the number line—represents a far larger infinity than the infinite set of natural numbers—the counting numbers. Readers learn to distinguish between countable infinity and uncountable infinity by way of a method known as a “one-to-one correspondence.” Mathematics students and enthusiasts are encouraged to proceed with care in both mathematics and life, lest they confuse countable infinity with uncountable infinity, large with unfathomably large, or order with disorder. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

scholarly journals On radicals of infinite matrix rings

1969 ◽  
Vol 16 (3) ◽  
pp. 195-203 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Infinite Matrix ◽  
Natural Numbers ◽  
Index Set ◽  
Matrix Rings ◽  
Infinite Set

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.

Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set

2017 ◽  
Vol 29 (4) ◽  
Author(s):  
Tiwadee Musunthia ◽  
Jörg Koppitz
Keyword(s):  
Natural Numbers ◽  
Maximal Subsemigroups ◽  
Countably Infinite ◽  
Infinite Set

AbstractIn this paper, we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively. We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers, we classify also all its maximal subsemigroups, containing a particular set of transformations.

scholarly journals Generalized cohesiveness

1999 ◽  
Vol 64 (2) ◽  
pp. 489-516 ◽  
Author(s):  
Tamara Hummel ◽  
Carl G. Jockusch
Keyword(s):  
Natural Numbers ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Infinite Set ◽  
Degrees Of Unsolvability ◽  
Computably Enumerable

AbstractWe study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show that for all n ≥ 2, there exists a n-cohesive set. We improve this result for n = 2 by showing that there is a 2-cohesive set. We show that the n-cohesive and n-r-cohesive degrees together form a linear, non-collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0(n+1) and also characterize the jumps of the n-r-cohesive degrees.

Independent Gödel sentences and independent sets

1975 ◽  
Vol 40 (2) ◽  
pp. 159-166
Author(s):  
A. M. Dawes ◽  
J. B. Florence
Keyword(s):  
Positive Integer ◽  
Independent Sets ◽  
Peano Arithmetic ◽  
Natural Numbers ◽  
Simple Set ◽  
Logical Notion ◽  
Infinite Set ◽  
Element Set

In this paper we investigate some of the recursion-theoretic problems which are suggested by the logical notion of independence.A set S of natural numbers will be said to be k-independent (respectively, ∞-independent) if, roughly speaking, in every correct system there is a k-element set (respectively, an infinite set) of independent true sentences of the form x ∈ S. S will be said to be effectively independent (respectively, absolutely independent) if given any correct system we can generate an infinite set of independent (respectively, absolutely independent) true sentences of the form x ∈ S.We prove that(a) S is absolutely independent ⇔S is effectively independent ⇔S is productive;(b) for every positive integer k there is a Π1 set which is k-independent but not (k + 1)-independent;(c) there is a Π1 set which is k-independent for all k but not ∞-independent;(d) there is a co-simple set which is ∞-independent.We also give two new proofs of the theorem of Myhill [1] on the existence of an infinite set of Σ1 sentences which are absolutely independent relative to Peano arithmetic. The first proof uses the existence of an absolutely independent Π1 set of natural numbers, and the second uses a modification of the method of Gödel and Rosser.

Theories incomparable with respect to relative interpretability

1962 ◽  
Vol 27 (2) ◽  
pp. 195-211 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Natural Numbers ◽  
Infinite Set

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

Ramsey's theorem and recursion theory

1972 ◽  
Vol 37 (2) ◽  
pp. 268-280 ◽  
Author(s):  
Carl G. Jockusch
Keyword(s):  
Recursion Theory ◽  
The Other ◽  
Natural Numbers ◽  
Arithmetical Hierarchy ◽  
Original Proof ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Infinite Sets ◽  
Recursive Partition ◽  
Positive Results

Let N be the set of natural numbers. If A ⊆ N, let [A]n denote the class of all n-element subsets of A. If P is a partition of [N]n into finitely many classes C1, …, Cp, let H(P) denote the class of those infinite sets A ⊆ N such that [A]n ⊆ Ci for some i. Ramsey's theorem [8, Theorem A] asserts that H(P) is nonempty for any such partition P. Our purpose here is to study what can be said about H(P) when P is recursive, i.e. each Ci, is recursive under a suitable coding of [N]n. We show that if P is such a recursive partition of [N]n, then H(P) contains a set which is Πn0 in the arithmetical hierarchy. In the other direction we prove that for each n ≥ 2 there is a recursive partition P of [N]n into two classes such that H(P) contains no Σn0 set. These results answer a question raised by Specker [12].A basic partition is a partition of [N]2 into two classes. In §§2, 3, and 4 we concentrate on basic partitions and in so doing prepare the way for the general results mentioned above. These are proved in §5. Our “positive” results are obtained by effectivizing proofs of Ramsey's theorem which differ from the original proof in [8]. We present these proofs (of which one is a generalization of the other) in §§4 and 5 in order to clarify the motivation of the effective versions.

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.

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