By using the notion of a simplified (κ,1)-morass, we construct κ-thin-tall, κ-thin-thick and, in a forcing extension, κ-very thin-thick superatomic Boolean algebras for every infinite regular cardinal κ.

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf = κ+. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra equation image such that equation image, equation image for every α < η and equation image. Especially, equation image and equation image can (...) be cardinal sequences of superatomic Boolean algebras. (shrink)

In connection with some known results on uncountable cardinal sequences for superatomic Boolean algebras, we shall describe some open questions for superatomic Boolean algebras concerning singular cardinals.

Let denote the class of all cardinal sequences of length α associated with compact scattered spaces. Also put If λ is a cardinal and α λ1>>λn−1 and ordinals α0,…,αn−1 such that α=α0++αn−1 and where each .The proofs are based on constructions of universal locally compact scattered spaces.

Let L be one of the topological languages L t , (L ∞ω ) t and (L κω ) t . We characterize the topological spaces which are models of the L-theory of the class of ordinals equipped with the order topology. The results show that the role played in classical model theory by the property of being well-ordered is taken over in the topological context by the property of being locally compact and scattered.

The analysis of the concept of orbital allows us to argue that—in opposition to a recent position in philosophy of science—it is impossible to defend the autonomy of the chemical reality in regard to physical reality, appealing to the idea that there is a conceptual rupture among a chemical interpretation and a quantuminterpretation of the concept. This is the case because there are not two different interpretations of the concept of orbital. On the contrary, the concept involved in structural chemistry (...) and in quantum chemistry comes from quantum mechanics.El análisis del concepto de orbital permite argumentar—en oposición a una propuesta reciente en filosofía de la química—que no es posible defender la autonomía ontológica de la realidad química respecto de la física, sobre la base de una ruptura conceptual entre una interpretación química del concepto y otra propia de la mecánica cuántica. Esto es así porque no existen tales interpretaciones distintas del concepto de orbital, sino que, por el contrario, este concepto en química cuántica y en química estructural proviene de la mecánica cuántica. (shrink)

In his recent Editorial Article, Jeffrey Seeman calls for the promotion of collaborative work among different disciplines, focusing on the case of the interaction between chemistry, the history of chemistry and the philosophy of chemistry. From a general viewpoint, it is difficult to disagree with this claim; moreover, the interest of scientists in the history and the philosophy of science is always welcome. However, the devil is in the details: there are several points that, we think, must be discussed more (...) carefully with the aim of arriving at far-reaching conclusions. (shrink)

Although during the last decades the philosophy of chemistry has greatly extended its thematic scope, the main difficulties appear in the attempt to link the chemical description of atoms and molecules and the description supplied by quantum mechanics. The aim of this paper is to analyze how the difficulties that threaten the continuous conceptual link between molecular chemistry and quantum mechanics can be overcome or, at least, moderated from the perspective of BM. With this purpose, in “The quantum-mechanical challenges” section (...) the foundational incompatibility between chemical and SQM descriptions will be briefly recalled. “Bohmian mechanics” section will be devoted to explain the main features of BM. In “Empirical equivalence and underdetermination” section, the consequences of the empirical equivalence between SQM and BM will be discussed. Finally, in the Conclusion, we will stress the scope of the obtained conclusions and the philosophical difficulties that still remain even after adopting BM for foundational purposes. (shrink)

We prove that if GCH holds and τ = 〈κα : α < η 〉 is a sequence of infinite cardinals such that κα ≥ |η | for each α < η, then there is a cardinal-preserving partial order that forces the existence of a scattered Boolean space whose cardinal sequence is τ.

We consider some questions of Donald Monk related to attainment of the tightness function in Boolean spaces. Then we prove in ZFC that attainment of tightness in the sense of the definition does not imply attainment of tightness in the free sequence sense. Our results give rise to full answers to [2, Problems 41, 42].

The countable sequences of cardinals which arise as cardinal sequences of superatomic Boolean algebras were characterized by La Grange on the basis of ZFC set theory. However, no similar characterization is available for uncountable cardinal sequences. In this paper we prove the following two consistency results:Ifθ = 〈κ α :α <ω 1〉 is a sequence of infinite cardinals, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that (...) θ is the cardinal sequence ofB.Ifκ is an uncountable cardinal such thatκ <κ =κ andθ = 〈κ α :α <κ +〉 is a cardinal sequence such thatκ α ≥κ for everyα <κ + andκ α =κ for everyα <κ + such that cf(α)<κ, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB. (shrink)