scholarly journals On the cofinality of ultrapowers

1999 ◽  
Vol 64 (2) ◽  
pp. 727-736 ◽  
Author(s):  
Andreas Blass ◽  
Heike Mildenberger
Keyword(s):  
Natural Numbers ◽  
Cardinal Characteristics ◽  
Splitting Number ◽  
The Continuum

AbstractWe prove some restrictions on the possible cofinalities of ultrapowers of the natural numbers with respect to ultrafilters on the natural numbers. The restrictions involve three cardinal characteristics of the continuum, the splitting number s, the unsplitting number r, and the groupwise density number g. We also prove some related results for reduced powers with respect to filters other than ultrafilters.

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.

scholarly journals Preservation of splitting families and cardinal characteristics of the continuum

2021 ◽  
Author(s):  
Martin Goldstern ◽  
Jakob Kellner ◽  
Diego A. Mejía ◽  
Saharon Shelah
Keyword(s):  
Finite Support ◽  
Cardinal Characteristics ◽  
Splitting Number ◽  
Cichoń’S Diagram ◽  
Dependent Entries ◽  
The Continuum

AbstractWe show how to construct, via forcing, splitting families that are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń’s diagram, $$\mathfrak{m}$$ m (2-Knaster), $$\mathfrak{p}$$ p , $$\mathfrak{h}$$ h , the splitting number $$\mathfrak{s}$$ s and the reaping number $$\mathfrak{r}$$ r .

Converse dual cardinals

2006 ◽  
Vol 71 (1) ◽  
pp. 22-34 ◽  
Author(s):  
Jörg Brendle ◽  
Shuguo Zhang
Keyword(s):  
Natural Numbers ◽  
Cardinal Invariants ◽  
The Continuum

AbstractWe investigate the set (ω) of partitions of the natural numbers ordered by ≤* where A ≤* B if by gluing finitely many blocks of A we can get a partition coarser than B. In particular, we determine the values of a number of cardinals which are naturally associated with the structure ((ω), ≥*), in terms of classical cardinal invariants of the continuum.

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)}.

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