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A poset hierarchy

Mirna Džamonja, Katherine Thompson (2006)

Open Mathematics

This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This...

Another proof of a result of Jech and Shelah

Péter Komjáth (2013)

Czechoslovak Mathematical Journal

Shelah’s pcf theory describes a certain structure which must exist if ω is strong limit and 2 ω > ω 1 holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that such partially...

Borel extensions of Baire measures in ZFC

Menachem Kojman, Henryk Michalewski (2011)

Fundamenta Mathematicae

We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.

Cellularity of free products of Boolean algebras (or topologies)

Saharon Shelah (2000)

Fundamenta Mathematicae

The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, θ = ( 2 c f ( μ ) ) + and 2 μ = μ + then there are Boolean algebras 𝔹 1 , 𝔹 2 such that c ( 𝔹 1 ) = μ , c ( 𝔹 2 ) < θ b u t c ( 𝔹 1 * 𝔹 2 ) = μ + . Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if 𝔹 is a ccc Boolean algebra and μ ω λ = c f ( λ ) 2 μ then 𝔹 satisfies the λ-Knaster condition (using the “revised GCH theorem”).

Continuous tree-like scales

James Cummings (2010)

Open Mathematics

Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at ℵω is consistent with Martin’s Maximum.

Embedding Cohen algebras using pcf theory

Saharon Shelah (2000)

Fundamenta Mathematicae

Using a theorem from pcf theory, we show that for any singular cardinal ν, the product of the Cohen forcing notions on κ, κ < ν, adds a generic for the Cohen forcing notion on ν + .

Equimorphism invariants for scattered linear orderings

Antonio Montalbán (2006)

Fundamenta Mathematicae

Two linear orderings are equimorphic if they can be embedded in each other. We define invariants for scattered linear orderings which classify them up to equimorphism. Essentially, these invariants are finite sequences of finite trees with ordinal labels. Also, for each ordinal α, we explicitly describe the finite set of minimal scattered equimorphism types of Hausdorff rank α. We compute the invariants of each of these minimal types..

Hybrid Prikry forcing

Dima Sinapova (2015)

Fundamenta Mathematicae

We present a new forcing notion combining diagonal supercompact Prikry forcing with interleaved extender based forcing. We start with a supercompact cardinal κ. In the final model the cofinality of κ is ω, the singular cardinal hypothesis fails at κ, and GCH holds below κ. Moreover we define a scale at κ which has a stationary set of bad points in the ground model.

Interpolation of κ -compactness and PCF

István Juhász, Zoltán Szentmiklóssy (2009)

Commentationes Mathematicae Universitatis Carolinae

We call a topological space κ -compact if every subset of size κ has a complete accumulation point in it. Let Φ ( μ , κ , λ ) denote the following statement: μ < κ < λ = cf ( λ ) and there is { S ξ : ξ < λ } [ κ ] μ such that | { ξ : | S ξ A | = μ } | < λ whenever A [ κ ] < κ . We show that if Φ ( μ , κ , λ ) holds and the space X is both μ -compact and λ -compact then X is κ -compact as well. Moreover, from PCF theory we deduce Φ ( cf ( κ ) , κ , κ + ) for every singular cardinal κ . As a corollary we get that a linearly Lindelöf and ω -compact space is uncountably compact, that is κ -compact for all uncountable cardinals κ .

Large cardinals and covering numbers

Pierre Matet (2009)

Fundamenta Mathematicae

The paper is concerned with the computation of covering numbers in the presence of large cardinals. In particular, we revisit Solovay's result that the Singular Cardinal Hypothesis holds above a strongly compact cardinal.

On splitting infinite-fold covers

Márton Elekes, Tamás Mátrai, Lajos Soukup (2011)

Fundamenta Mathematicae

Let X be a set, κ be a cardinal number and let ℋ be a family of subsets of X which covers each x ∈ X at least κ-fold. What assumptions can ensure that ℋ can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover ℋ: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ℝⁿ by polyhedra and by arbitrary convex sets. We focus on...

Pcf theory and cardinal invariants of the reals

Lajos Soukup (2011)

Commentationes Mathematicae Universitatis Carolinae

The additivity spectrum ADD ( ) of an ideal 𝒫 ( I ) is the set of all regular cardinals κ such that there is an increasing chain { A α : α < κ } with α < κ A α . We investigate which set A of regular cardinals can be the additivity spectrum of certain ideals. Assume that = or = 𝒩 , where denotes the σ -ideal generated by the compact subsets of the Baire space ω ω , and 𝒩 is the ideal of the null sets. We show that if A is a non-empty progressive set of uncountable regular cardinals and pcf ( A ) = A , then ADD ( ) = A in some c.c.c generic extension of the...

Reflection implies the SCH

Saharon Shelah (2008)

Fundamenta Mathematicae

We prove that, e.g., if μ > cf(μ) = ℵ₀ and μ > 2 and every stationary family of countable subsets of μ⁺ reflects in some subset of μ⁺ of cardinality ℵ₁, then the SCH for μ⁺ holds (moreover, for μ⁺, any scale for μ⁺ has a bad stationary set of cofinality ℵ₁). This answers a question of Foreman and Todorčević who get such a conclusion from the simultaneous reflection of four stationary sets.

Strong covering without squares

Saharon Shelah (2000)

Fundamenta Mathematicae

Let W be an inner model of ZFC. Let κ be a cardinal in V. We say that κ-covering holds between V and W iff for all X ∈ V with X ⊆ ON and V ⊨ |X| < κ, there exists Y ∈ W such that X ⊆ Y ⊆ ON and V ⊨ |Y| < κ. Strong κ-covering holds between V and W iff for every structure M ∈ V for some countable first-order language whose underlying set is some ordinal λ, and every X ∈ V with X ⊆ λ and V ⊨ |X| < κ, there is Y ∈ W such that X ⊆ Y ≺ M and V ⊨ |Y| < κ.   We prove that if κ is V-regular,...

Well-quasi-ordering Aronszajn lines

Carlos Martinez-Ranero (2011)

Fundamenta Mathematicae

We show that, assuming PFA, the class of all Aronszajn lines is well-quasi-ordered by embeddability.

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