Monotone reducibility and the family of infinite sets

1984 ◽  
Vol 49 (3) ◽  
pp. 774-782 ◽  
Author(s):  
Douglas Cenzer
Keyword(s):  
Natural Numbers ◽  
Arithmetical Hierarchy ◽  
Continuous Map ◽  
Borel Hierarchy ◽  
The Family ◽  
Wadge Reducibility ◽  
Borel Map ◽  
Infinite Sets ◽  
Compact Subsets

AbstractLet A and B be subsets of the space 2N of sets of natural numbers. A is said to be Wadge reducible to B if there is a continuous map Φ from 2N into 2N such that A = Φ−1 (B); A is said to be monotone reducible to B if in addition the map Φ is monotone, that is, a ⊂ b implies Φ(a) ⊂ Φ(b). The set A is said to be monotone if a ∈ A and a ⊂ b imply b ∈ A. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The sets are all reducible to the ( but not ) sets, which are in turn all reducible to the strictly sets, which are all in turn reducible to the strictly sets. In addition, the nontrivial sets all have the same degree for n ≤ 2. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly monotone sets which have different monotone degrees. We show that every monotone set is actually positive. We also consider reducibility for subsets of the space of compact subsets of 2N. This leads to the result that the finitely iterated Cantor-Bendixson derivative Dn is a Borel map of class exactly 2n, which answers a question of Kuratowski.

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.

scholarly journals On the set of finite subsets of a set

1975 ◽  
Vol 20 (1) ◽  
pp. 38-45
Author(s):  
J. L Hickman
Keyword(s):  
Natural Numbers ◽  
Finite Sets ◽  
Infinite Sets

We sometimes think of medial (that is, infinite Dedekind-finite) sets as being “small” infinite sets. Medial cardinals can be defined as those cardinals that are incomparable to ℵℴ; hence we tend to think of them as being spread out on a plane “just above” the natural numbers, which seems to lend support to the view expressed above that medial sets are “small”.

scholarly journals Li-Yorke Sensitivity of Set-Valued Discrete Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Heng Liu ◽  
Fengchun Lei ◽  
Lidong Wang
Keyword(s):  
Metric Space ◽  
Short Proof ◽  
Hausdorff Metric ◽  
Discrete Systems ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonempty Compact ◽  
Compact Subsets

Consider the surjective, continuous mapf:X→Xand the continuous mapf¯of𝒦(X)induced byf, whereXis a compact metric space and𝒦(X)is the space of all nonempty compact subsets ofXendowed with the Hausdorff metric. In this paper, we give a short proof that iff¯is Li-Yoke sensitive, thenfis Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity offdoes not imply Li-Yorke sensitivity off¯.

scholarly journals On a Sumset Conjecture of Erdős

2015 ◽  
Vol 67 (4) ◽  
pp. 795-809 ◽  
Author(s):  
Mauro Di Nasso ◽  
Isaac Goldbring ◽  
Renling Jin ◽  
Steven Leth ◽  
Martino Lupini ◽  
...  
Keyword(s):  
Positive Solution ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Amenable Groups ◽  
Banach Density ◽  
Infinite Sets ◽  
Erdös Conjecture

AbstractErdős conjectured that for any set A ⊆ ℕ with positive lower asymptotic density, there are infinite sets B;C ⊆ ℕ such that B + C ⊆ A. We verify Erdős’ conjecture in the case where A has Banach density exceeding ½ . As a consequence, we prove that, for A ⊆ ℕ with positive Banach density (amuch weaker assumption than positive lower density), we can find infinite B;C ⊆ ℕ such that B+C is contained in the union of A and a translate of A. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.

Sets which do not have subsets of every higher degree

1978 ◽  
Vol 43 (1) ◽  
pp. 135-138 ◽  
Author(s):  
Stephen G. Simpson
Keyword(s):  
Similar Degree ◽  
Theoretic Analysis ◽  
Natural Numbers ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Infinite Sets ◽  
Special Case

Let A be a subset of ω, the set of natural numbers. The degree of A is its degree of recursive unsolvability. We say that A is rich if every degree above that of A is represented by a subset of A. We say that A is poor if no degree strictly above that of A is represented by a subset of A. The existence of infinite poor (and hence nonrich) sets was proved by Soare [9].Theorem 1. Suppose that A is infinite and not rich. Then every hyperarith-metical subset H of ω is recursive in A.In the special case when H is arithmetical, Theorem 1 was proved by Jockusch [4] who employed a degree-theoretic analysis of Ramsey's theorem [3]. In our proof of Theorem 1 we employ a similar, degree-theoretic analysis of a certain generalization of Ramsey's theorem. The generalization of Ramsey's theorem is due to Nash-Williams [6]. If A ⊆ ω we write [A]ω for the set of all infinite subsets of A. If P ⊆ [ω]ω we let H(P) be the set of all infinite sets A such that either [A]ω ⊆ P = ∅. Nash-Williams' theorem is essentially the statement that if P ⊆ [ω]ω is clopen (in the usual, Baire topology on [ω]ω) then H(P) is nonempty. Subsequent, further generalizations of Ramsey's theorem were proved by Galvin and Prikry [1], Silver [8], Mathias [5], and analyzed degree-theoretically by Solovay [10]; those results are not needed for this paper.

Frege's theorem in a constructive setting

1999 ◽  
Vol 64 (2) ◽  
pp. 486-488 ◽  
Author(s):  
John L. Bell
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
The Family ◽  
Power Set ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Image Position ◽  
Excluded Middle ◽  
Intuitionistic Set Theory

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map υ from the power set of E to E satisfying the conditionthen E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in Section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map υ be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory, the followingTheorem. Let υ be a map with domain a family of subsets of a set E to E satisfying the following conditions:(i) ø ϵdom(υ)(ii)∀U ϵdom(υ)∀x ϵ E − UU ∪ x ϵdom(υ)(iii)∀UV ϵdom(5) υ(U) = υ(V) ⇔ U ≈ V.Then we can define a subset N of E which is the domain of a model of Peano's axioms.

Inconsistency of uncountable infinite sets under ZFC framework

Kybernetes ◽  
2008 ◽  
Vol 37 (3/4) ◽  
pp. 453-457 ◽  
Author(s):  
Wujia Zhu ◽  
Yi Lin ◽  
Guoping Du ◽  
Ningsheng Gong
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Design Methodology ◽  
Natural Numbers ◽  
Real Numbers ◽  
Conceptual Approach ◽  
Content Type ◽  
Infinite Sets ◽  
Axiomatic Set Theory ◽  
First Time

PurposeThe purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsGiven the fact that the set N={x|n(x)} of all natural numbers, where n(x)=df “x is a natural number” is a self‐contradicting non‐set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non‐existent or self‐contradicting non‐sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self‐contradicting non‐set.Originality/valueThe first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

Sequential discreteness and clopen-I-Boolean classes

1987 ◽  
Vol 52 (1) ◽  
pp. 232-242
Author(s):  
Randall Dougherty
Keyword(s):  
Final Section ◽  
Baire Space ◽  
Boolean Operations ◽  
Index Set ◽  
Borel Sets ◽  
Borel Hierarchy ◽  
Wadge Reducibility ◽  
Analytic Sets ◽  
Converse Results

Kantorovich and Livenson [6] initiated the study of infinitary Boolean operations applied to the subsets of the Baire space and related spaces. It turns out that a number of interesting collections of subsets of the Baire space, such as the collection of Borel sets of a given type (e.g. the Fσ sets) or the collection of analytic sets, can be expressed as the range of an ω-ary Boolean operation applied to all possible ω-sequences of clopen sets. (Such collections are called clopen-ω-Boolean.) More recently, the ranges of I-ary Boolean operations for uncountable I have been considered; specific questions include whether the collection of Borel sets, or the collection of sets at finite levels in the Borel hierarchy, is clopen-I-Boolean.The main purpose of this paper is to give a characterization of those collections of subsets of the Baire space (or similar spaces) that are clopen-I-Boolean for some I. The Baire space version can be stated as follows: a collection of subsets of the Baire space is clopen-I-Boolean for some I iff it is nonempty and closed downward and σ-directed upward under Wadge reducibility, and in this case we may take I = ω2. The basic method of proof is to use discrete subsets of spaces of the form K2 to put a number of smaller clopen-I-Boolean classes together to form a large one. The final section of the paper gives converse results indicating that, at least in some cases, ω2 cannot be replaced by a smaller index set.

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