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Characterizing the powerset by a complete (Scott) sentence

Ioannis Souldatos (2013)

Fundamenta Mathematicae

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence ϕ ℳ if ϕ ℳ has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if ℵ β is characterized by a Scott sentence, then 2 ℵ β + β ₁ is (homogeneously) characterized by a Scott sentence, for all 0 < β₁ < ω₁....

Compacta are maximally G δ -resolvable

István Juhász, Zoltán Szentmiklóssy (2013)

Commentationes Mathematicae Universitatis Carolinae

It is well-known that compacta (i.e. compact Hausdorff spaces) are maximally resolvable, that is every compactum X contains Δ ( X ) many pairwise disjoint dense subsets, where Δ ( X ) denotes the minimum size of a non-empty open set in X . The aim of this note is to prove the following analogous result: Every compactum X contains Δ δ ( X ) many pairwise disjoint G δ -dense subsets, where Δ δ ( X ) denotes the minimum size of a non-empty G δ set in X .

Decidability and definability results related to the elementary theory of ordinal multiplication

Alexis Bès (2002)

Fundamenta Mathematicae

The elementary theory of ⟨α;×⟩, where α is an ordinal and × denotes ordinal multiplication, is decidable if and only if α < ω ω . Moreover if | r and | l respectively denote the right- and left-hand divisibility relation, we show that Th ⟨ ω ω ξ ; | r ⟩ and Th ⟨ ω ξ ; | l ⟩ are decidable for every ordinal ξ. Further related definability results are also presented.

Families of almost disjoint Hamel bases.

Lorenz Halbeisen (2005)

Extracta Mathematicae

For infinite dimensional Banach spaces X we investigate the maximal size of a family of pairwise almost disjoint normalized Hamel bases of X, where two sets A and B are said to be almost disjoint if the cardinality of A ∩ B is smaller than the cardinality of either A or B.

Fields of surreal numbers and exponentiation

Lou van den Dries, Philip Ehrlich (2001)

Fundamenta Mathematicae

We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case,...

Hausdorff ’s theorem for posets that satisfy the finite antichain property

Uri Abraham, Robert Bonnet (1999)

Fundamenta Mathematicae

Hausdorff characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulate and prove a corresponding characterization of the class of scattered partial orderings that satisfy the finite antichain condition (FAC).  Consider the least class of partial orderings containing the class of well-founded orderings that satisfy the FAC and is closed under the following operations: (1)...

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