Artículos con la etiqueta ‘teoría de conjuntos’

Arrow’s Theorem by Arrow Theory

Por • 21 ene, 2014 • Category: Filosofía

We give a categorical account of Arrow’s theorem, a seminal result in social choice theory.

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.

WISC may fail in the category of sets

Por • 22 nov, 2013 • Category: Crítica

By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos due to Shulman, we show that the very weak choice principle WISC—the statement that for every set there is a set of surjections to it cofinal in all such surjections—is independent of the rest of the axioms of the categorical constructive set theory given by a well-pointed topos.

The Foundations of Mathematics in the Physical Reality

Por • 16 nov, 2013 • Category: Crítica

The mathematical concept of a set can be used as the foundation for all known mathematics, and the assertions known as the Peano axioms for the set of all natural numbers are used to be considered as the fountainhead of all mathematical knowledge. In «The Foundations of Mathematics in the Physical Reality» is given an axiomatic definition of sets and of the natural numbers. As close as possible are followed the common axioms used in the literature. But there are big differences.

Husserl, Cantor & Hilbert: La Grande Crise des Fondements Mathematiques du XIXeme Siecle

Por • 13 nov, 2013 • Category: Educacion

Three thinkers of the 19th century revolutionized the science of logic, mathematics, and philosophy. Edmund Husserl (1859-1938), mathematician and a disciple of Karl Weierstrass, made an immense contribution to the theory of human thought. The paper offers a complex analysis of Husserl’s mathematical writings covering calculus of variations, differential geometry, and theory of numbers which laid the ground for his later phenomenological breakthrough. Georg Cantor (1845-1818), the creator of set theory, was a mathematician who changed the mathematical thinking per se. By analyzing the philosophy of set theory this paper shows how was it possible (by introducing into mathematics what philosophers call ‘the subject’). Set theory happened to be the most radical answer to the crisis of foundations. David Hilbert (1862-1943), facing the same foundational crisis, came up with his axiomatic method, indeed a minimalist program whose roots can be traced back to Descartes and Cauchy. Bringing together these three key authors, the paper is the first attempt to analyze how the united efforts of philosophy and mathematics helped to dissolve the epistemological crisis of the 19th century.

A topological set theory implied by ZF and GPK+∞

Por • 29 sep, 2013 • Category: Opinion

We present a system of axioms motivated by a topological intuition: The set of subsets of any set is a topology on that set. On the one hand, this system is acommon weakening of Zermelo-Franenkel set theory ZF, the positive set theory GPK and he theory of hyperuniverses. On the other hand, it retains most of the expressiveness of theses theories and has the same consistency strength as ZF. We single out the additional axiom of the universal set as the one that increases the consistency strength to that of GPK and explore several other axioms and interrlations between those theories. Our results are independent of whether the empty class is a set and whether atoms exist.

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 $B_n$.

Constructive Zermelo-Fraenkel set theory and the limited principle of omniscience

Por • 15 feb, 2013 • Category: Crítica

In recent years the question of whether adding the limited principle of omniscience, LPO, to constructive Zermelo-Fraenkel set theory, CZF, increases its strength has arisen several times. As the addition of excluded middle for atomic formulae to CZF results in a rather strong theory, i.e. much stronger than classical Zermelo set theory, it is not obvious that its augmentation by LPO would be proof-theoretically benign. The purpose of this paper is to show that CZF +RDC+ LPO has indeed the same strength as CZF, where RDC stands for relativized dependent choice. In particular, these theories prove the same Pi-?0-2 theorems of arithmetic.

Alain Badiou’s Mistake — Two Postulates of Dialectic Materialism

Por • 8 ene, 2013 • Category: Filosofía

To accompany recent openings in category theory and philosophy, I discuss how Alain Badiou attempts to rephrase his dialectic philosophy in topos-theoretic terms. Topos theory bridges the problems emerging in set-theoretic language by a categorical approach that reinscribes set-theoretic language in a categorical framework. Badiou’s own topos-theoretic formalism, however, turns out to be confined only to a limited, set-theoretically bounded branch of locales. This results with his reduced mathematical understanding of the ‘postulate of materialism’ constitutive to his account. Badiou falsely assumes this postulate to be singular whereas topos theory reveals its two-sided nature whose synthesis emerges only as a result of a (quasi-)split structure of truth. Badiou thus struggles with his own mathematical argument. I accomplish a correct version of his proof the sets defined over such a ‘transcendental algebra’ T form a (local) topos. Finally, I discuss the philosophical implications Badiou’s mathematical inadequacies entail.

A multiverse perspective on the axiom of constructiblity

Por • 30 oct, 2012 • Category: Filosofía

I shall argue that the commonly held V not equal L via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V = L inside larger set-theoretic universes.