Cardinal reflections and point-character of uniformities – counterexamples
We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most levels of size ω. We also give a complete ZFC characterization of the cardinal sequences of regular scattered spaces. Although the classes of regular and of 0-dimensional scattered spaces are different, we prove that they have the same cardinal sequences.
Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put . We show that f ∈ (α) iff for some natural number n there are infinite cardinals and ordinals such that and where each . Under GCH we prove that if α < ω₂ then (i) ; (ii) if λ > cf(λ) = ω, ; (iii) if cf(λ) = ω₁, ; (iv) if cf(λ) > ω₁, . This yields a complete characterization of the classes (α) for all α < ω₂,...
The main purpose of this article is to introduce the categorical concept of pullback in Mizar. In the first part of this article we redefine homsets, monomorphisms, epimorpshisms and isomorphisms [7] within a free-object category [1] and it is shown there that ordinal numbers can be considered as categories. Then the pullback is introduced in terms of its universal property and the Pullback Lemma is formalized [15]. In the last part of the article we formalize the pullback of functors [14] and it...
We continue the work of [2] and prove that for λ successor, a λ-categorical theory T in is μ-categorical for every μ ≤ λ which is above the -beth cardinal.
We assume a theory T in the logic is categorical in a cardinal λ κ, and κ is a measurable cardinal. We prove that the class of models of T of cardinality < λ (but ≥ |T|+κ) has the amalgamation property; this is a step toward understanding the character of such classes of models.
We study categoricity in power for reduced models of first order logic without equality.