Combinatorics for the dominating and unsplitting numbers

2004 ◽  
Vol 69 (2) ◽  
pp. 482-498 ◽  
Author(s):  
Jason Aubrey
Keyword(s):  
Baire Space ◽  
Minimum Cardinality ◽  
Cardinal Characteristics ◽  
The Continuum

Abstract.In this paper we introduce a new property of families of functions on the Baire space, called pseudo-dominating, and apply the properties of these families to the study of cardinal characteristics of the continuum. We show that the minimum cardinality of a pseudo-dominating family is min {τ, ∂}. We derive two corollaries from the proof: τ ≥ min{∂, u} and min{∂, τ} = min{∂, τσ}. We show that if a dominating family is partitioned into fewer that s pieces, then one of the pieces is pseudo-dominating. We finally show that u < g implies that every unbounded family of functions is pseudo-dominating, and that the Filter Dichotomy principle is equivalent to every unbounded family of functions being finitely pseudo-dominating.

Monotone inductive definitions over the continuum

1976 ◽  
Vol 41 (1) ◽  
pp. 188-198 ◽  
Author(s):  
Douglas Cenzer
Keyword(s):  
Mathematical Logic ◽  
Recursion Theory ◽  
Axiom System ◽  
Baire Space ◽  
Natural Numbers ◽  
Inductive Definition ◽  
Inductive Definitions ◽  
Definition Of ◽  
The Continuum

Monotone inductive definitions occur frequently throughout mathematical logic. The set of formulas in a given language and the set of consequences of a given axiom system are examples of (monotone) inductively defined sets. The class of Borel subsets of the continuum can be given by a monotone inductive definition. Kleene's inductive definition of recursion in a higher type functional (see [6]) is fundamental to modern recursion theory; we make use of it in §2.Inductive definitions over the natural numbers have been studied extensively, beginning with Spector [11]. We list some of the results of that study in §1 for comparison with our new results on inductive definitions over the continuum. Note that for our purposes the continuum is identified with the Baire space ωω.It is possible to obtain simple inductive definitions over the continuum by introducing real parameters into inductive definitions over N—as in the definition of recursion in [5]. This is itself an interesting concept and is discussed further in [4]. These parametric inductive definitions, however, are in general weaker than the unrestricted set of inductive definitions, as is indicated below.In this paper we outline, for several classes of monotone inductive definitions over the continuum, solutions to the following characterization problems:(1) What is the class of sets which may be given by such inductive definitions ?(2) What is the class of ordinals which are the lengths of such inductive definitions ?These questions are made more precise below. Most of the results of this paper were announced in [2].

Simple Cardinal Characteristics of the Continuum

2002 ◽  
Vol 8 (4) ◽  
pp. 552
Author(s):  
Heike Mildenberger ◽  
Andreas Blass ◽  
Haim Judah
Keyword(s):  
Cardinal Characteristics ◽  
The Continuum

scholarly journals Cardinal characteristics of the continuum and partitions

2019 ◽  
Vol 235 (1) ◽  
pp. 13-38
Author(s):  
William Chen ◽  
Shimon Garti ◽  
Thilo Weinert
Keyword(s):  
Cardinal Characteristics ◽  
The Continuum

scholarly journals Hereditary interval algebras and cardinal characteristics of the continuum

2021 ◽  
Author(s):  
Michael Hrušák ◽  
Carlos Azarel Martínez-Ranero ◽  
Ulises Ariet Ramos-García
Keyword(s):  
Cardinal Characteristics ◽  
The Continuum

scholarly journals σ-Continuous Functions and Related Cardinal Characteristics of the Continuum

2020 ◽  
Vol 76 (1) ◽  
pp. 1-10
Author(s):  
Taras Banakh
Keyword(s):  
Continuous Functions ◽  
Topological Spaces ◽  
Cardinal Characteristics ◽  
The Continuum

AbstractA function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous) if there exists a (closed) cover {Xn}n∈ω of X such that for every n ∈ ω the restriction f ↾ Xn is continuous. By 𝔠 σ (resp. 𝔠¯σ)we denote the largest cardinal κ ≤ 𝔠 such that every function f : X → ℝ defined on a subset X ⊂ ℝ of cardinality |X| <κ is σ-continuous (resp. ¯σ-continuous). It is clear that ω1 ≤ 𝔠¯σ ≤ 𝔠 σ ≤ 𝔠.We prove that 𝔭 ≤ 𝔮0 = 𝔠¯σ =min{𝔠 σ, 𝔟, 𝔮 }≤ 𝔠 σ ≤ min{non(ℳ), non(𝒩)}.

scholarly journals THE POLARISED PARTITION RELATION FOR ORDER TYPES

2020 ◽  
Vol 71 (3) ◽  
pp. 823-842
Author(s):  
L D Klausner ◽  
T Weinert
Keyword(s):  
Natural Numbers ◽  
Partition Relation ◽  
Cardinal Characteristics ◽  
Order Types ◽  
The Given ◽  
The Continuum

Abstract We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal characteristics of the continuum. We build on work by Erd̋s, Garti, Jones, Orr, Rado, Shelah and Szemerédi. In particular, we show that a theorem of Jones extends from the natural numbers to the rational ones, but consistently extends only to three further equimorphism classes of countable orderings. This is made possible by applying a 13-year-old theorem of Orr about embedding a given order into a sum of finite orders indexed over the given order.

Non-constructive Galois-Tukey connections

1997 ◽  
Vol 62 (4) ◽  
pp. 1179-1186 ◽  
Author(s):  
Heike Mildenberger
Keyword(s):  
Cardinal Characteristics ◽  
The Continuum

AbstractThere are inequalities between cardinal characteristics of the continuum that are true in any model of ZFC, but without a Borel morphism proving the inequality. We answer some questions from Blass [1].

Sign in / Sign up

Export Citation Format

Share Document