Artículos con la etiqueta ‘continuo matemático’

A cognitive analysis of Cauchy’s conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus

Por • 12 ene, 2014 • Category: Opinion

In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It highlights the conceptual power of Cauchy’s vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern set-theoretic formulation of mathematical analysis.



On ground model definability

Por • 6 dic, 2013 • Category: Opinion

Laver, and Woodin independently, showed that models of ZFC are uniformly definable in their set-forcing extensions, using a ground model parameter. We investigate ground model definability for models of fragments of ZFC , particularly of ZF+DC δ and of ZFC − , and we obtain both positive and negative results. Generalizing the results of Laver and Woodin, we show that models of ZF+DC δ are uniformly definable in their set-forcing extensions by posets admitting a gap at δ , using a ground model parameter. In particular, this means that models of ZF+DC δ are uniformly definable in their forcing extensions by posets of size less than δ . We also show that it is consistent for ground model definability to fail for models of ZFC − of the form H κ + . Using forcing, we produce a ZFC universe in which there is a cardinal κ>>ω such that H κ + is not definable in its Cohen forcing extension. As a corollary, we show that there is always a countable transitive model of ZFC − violating ground model definability. These results turn out to have a bearing on ground model definability for models of ZFC . It follows from our proof methods that the hereditary size of the parameter that Woodin used to define a ZFC model in its set-forcing extension is best possible.



Title: Reduced products of UHF algebras under forcing axioms

Por • 25 mar, 2013 • Category: Crítica

If $A_n$ is a sequence of C*-algebras, then the C*-algebra $\prod A_n / \bigoplus A_n$ is called a reduced product. We prove, assuming Todorcevic’s Axiom and Martin’s Axiom, that every isomorphism between two reduced products of separable, unital UHF algebras must be definable in a strong sense. As a corollary we deduce that two such reduced products $\prod A_n / \bigoplus A_n$ and $\prod B_n / \bigoplus B_n$ are isomorphic if and only if, up to an almost-permutation of $\mathbb{N}$, $A_n$ is isomorphic to $B_n$.