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.

Ultrafilters on the natural numbers

2003 ◽  
Vol 68 (3) ◽  
pp. 764-784 ◽  
Author(s):  
Christopher Barney
Keyword(s):  
Partial Characterization ◽  
Natural Numbers ◽  
Cardinal Invariant ◽  
Compact Closure ◽  
Cardinal Invariants ◽  
The Core ◽  
Generic Existence ◽  
Invariant Equality ◽  
The Continuum

AbstractWe study the problem of existence and generic existence of ultrafilters on ω. We prove a conjecture of Jörg Brendle's showing that there is an ultrafilter that is countably closed but is not an ordinal ultrafilter under CH. We also show that Canjar's previous partial characterization of the generic existence of Q-points is the best that can be done. More simply put, there is no normal cardinal invariant equality that fully characterizes the generic existence of Q-points. We then sharpen results on generic existence with the introduction of σ-compact ultrafilters. We show that the generic existence of said ultrafilters is equivalent to . This result, taken along with our result that there exists a Kσ, non-countably closed ultrafilter under CH, expands the size of the class of ultrafilters that were known to fit this description before. From the core of the proof, we get a new result on the cardinal invariants of the continuum, i.e., the cofinality of the sets with σ-compact closure is .

Almost Disjoint Families and Diagonalizations of Length Continuum

2010 ◽  
Vol 16 (2) ◽  
pp. 240-260
Author(s):  
Dilip Raghavan
Keyword(s):  
Cardinal Invariants ◽  
Almost Disjoint ◽  
Almost Disjoint Families ◽  
The Continuum

AbstractWe present a survey of some results and problems concerning constructions which require a diagonalization of length continuum to be carried out, particularly constructions of almost disjoint families of various sorts. We emphasize the role of cardinal invariants of the continuum and their combinatorial characterizations in such constructions.

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].

A Comment on “ 𝔭 < t”

2009 ◽  
Vol 52 (2) ◽  
pp. 303-314 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Weak Version ◽  
Cardinal Invariants ◽  
The Continuum

AbstractDealing with the cardinal invariants 𝔭 and t of the continuum, we prove that m = p = ℵ2 ⇒ t = ℵ2. In other words, if MAℵ1 (or a weak version of this) holds, then (of course ℵ2 ≤ 𝔭 ≤ t and) 𝔭 = ℵ2 ⇒ 𝔭 = t. The proof is based on a criterion for 𝔭 < t.

scholarly journals Cardinal invariants above the continuum

1995 ◽  
Vol 75 (3) ◽  
pp. 251-268 ◽  
Author(s):  
James Cummings ◽  
Saharon Shelah
Keyword(s):  
Cardinal Invariants ◽  
The Continuum

scholarly journals ISOLATING CARDINAL INVARIANTS

2003 ◽  
Vol 03 (01) ◽  
pp. 143-162 ◽  
Author(s):  
JINDŘICH ZAPLETAL
Keyword(s):  
Cardinal Invariants ◽  
The Continuum

There is an optimal way of increasing certain cardinal invariants of the continuum.

scholarly journals After all, there are some inequalities which are provable in ZFC

2000 ◽  
Vol 65 (2) ◽  
pp. 803-816
Author(s):  
Tomek Bartoszyński ◽  
Andrzej Rosłanowski ◽  
Saharon Shelah
Keyword(s):  
Cardinal Invariants ◽  
The Continuum

AbstractWe address ZFC inequalities between some cardinal invariants of the continuum, which turned out to be true in spite of strong expectations given by [11].

Some initial segments of the Rudin-Keisler ordering

1981 ◽  
Vol 46 (1) ◽  
pp. 147-157 ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Continuum Hypothesis ◽  
Natural Numbers ◽  
Isomorphism Classes ◽  
The Continuum

AbstractA 2-affable ultrafilter has only finitely many predecessors in the Rudin-Keisler ordering of isomorphism classes of ultrafilters over the natural numbers. If the continuum hypothesis is true, then there is an ℵ1-sequence of ultrafilters Dα such that the strict Rudin-Keisler predecessors of Dα are precisely the isomorphs of the Dβ's for β < α.

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