Zeno's Paradoxes. A Cardinal Problem 1. On Zenonian Plurality

In this paper the claim that Zeno's paradoxes have been solved is contested. Although no one has ever touched Zeno without refuting him (Whitehead), it will be our aim to show that, whatever it was that was refuted, it was certainly not Zeno. The paper is organised in two parts. In the first part we will demonstrate that upon direct analysis of the Greek sources, an underlying structure common to both the Paradoxes of Plurality and the Paradoxes of Motion can be exposed. This structure bears on a correct - Zenonian - interpretation of the concept of division through and through. The key feature, generally overlooked but essential to a correct understanding of all his arguments, is that they do not presuppose time. Division takes place simultaneously. This holds true for both PP and PM. In the second part a mathematical representation will be set up that catches this common structure, hence the essence of all Zeno's arguments, however without refuting them. Its central tenet is an aequivalence proof for Zeno's procedure and Cantor's Continuum Hypothesis. Some number theoretic and geometric implications will be shortly discussed. Furthermore, it will be shown how the Received View on the motion-arguments can easely be derived by the introduction of time as a (non-Zenonian) premiss, thus causing their collapse into arguments which can be approached and refuted by Aristotle's limit-like concept of the potentially infinite, which remained - though in different disguises - at the core of the refutational strategies that have been in use up to the present. Finally, an interesting link to Newtonian mechanics via Cremona geometry can be established.Comment: 41 pages, 7 figure

Contributions to the theory of Large Cardinals through the method of Forcing

[eng] The present dissertation is a contribution to the field of Mathematical Logic and, more particularly, to the subfield of Set Theory. Within Set theory, we are mainly concerned with the interactions between the largecardinal axioms and the method of Forcing. This is the line of research with a deeper impact in the subsequent configuration of modern Mathematics. This area has found many central applications in Topology [ST71][Tod89], Algebra [She74][MS94][DG85][Dug85], Analysis [Sol70] or Category Theory [AR94][Bag+15], among others. The dissertation is divided in two thematic blocks: In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopenka’s Principle (Part I). In Block II we make a contribution to Singular Cardinal Combinatorics (Part II and Part III). Specifically, in Part I we investigate the Identity Crisis phenomenon in the region comprised between the first supercompact cardinal and Vopenka’s Principle. As a result, we settle all the questions that were left open in [Bag12, §5]. Afterwards, we present a general theory of preservation of C(n)– extendible cardinals under class forcing iterations from which we derive many applications. In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis (SCH) and other combinatorial principles, such as the tree property or the reflection of stationary sets. In Part II we generalize the main theorems of [FHS18] and [Sin16] and manage to weaken the largecardinal hypotheses necessary for Magidor-Shelah’s theorem [MS96]. Finally, in Part III we introduce the concept of _-Prikry forcing as a generalization of the classical notion of Prikry-type forcing. Subsequently we devise an abstract iteration scheme for this family of posets and, as an application, we prove the consistency of ZFC + ¬SCH_ + Refl([cat] La present tesi és una contribució a l’estudi de la Lògica Matemàtica i més particularment a la Teoria de Conjunts. Dins de la Teoria de Conjunts, la nostra àrea de recerca s’emmarca dins l’estudi de les interaccions entre els Axiomes de Grans Cardinals i el mètode de Forcing. Aquestes dues eines han tigut un impacte molt profund en la configuració de la matemàtica contemporànea com a conseqüència de la resolució de qüestions centrals en Topologia [ST71][Tod89], Àlgebra [She74][MS94][DG85][Dug85], Anàlisi Matemàtica [Sol70] o Teoria de Categories [AR94][Bag+15], entre d’altres. La tesi s’articula entorn a dos blocs temàtics. Al Bloc I analitzem la jerarquia de Grans Cardinals compresa entre el primer cardinal supercompacte i el Principi de Vopenka (Part I), mentre que al Bloc II estudiem alguns problemes de la Combinatòria Cardinal Singular (Part II i Part III). Més precisament, a la Part I investiguem el fenòmen de Crisi d’Identitat en la regió compresa entre el primer cardinal supercompacte i el Principi de Vopenka. Com a conseqüència d’aquesta anàlisi resolem totes les preguntes obertes de [Bag12, §5]. Posteriorment presentem una teoria general de preservació de cardinals C(n)–extensibles sota iteracions de longitud ORD, de la qual en derivem nombroses aplicacions. A la Part II i Part III analitzem la relació entre la Hipòtesi dels Cardinals Singulars (SCH) i altres principis combinatoris, tals com la Propietat de l’Arbre o la reflexió de conjunts estacionaris. A la Part II obtenim sengles generalitzacions dels teoremes principals de [FHS18] i [Sin16] i afeblim les hipòtesis necessàries perquè el teorema de Magidor-Shelah [MS96] siga cert. Finalment, a la Part III, introduïm el concepte de forcing _-Prikry com a generalització de la noció clàssica de forcing del tipus Prikry. Posteriorment dissenyem un esquema d’iteracions abstracte per aquesta família de forcings i, com a aplicació, derivem la consistència de ZFC + ¬SCH_ + Refl

Set Theory

This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C*-algebras, Ramsey theory, measure theory, representation theory, group theory and Banach spaces