Artículos con la etiqueta ‘teoría de conjuntos’

The fundamentals of relations language mathematics

Por • 23 oct, 2012 • Category: Filosofía

The fundamentals of formal logic, theory of sets and mathematical structures are narrated in terms of relations language.



Axiomatic Method and Category Theory

Por • 5 oct, 2012 • Category: Filosofía

Lawvere’s axiomatization of topos theory and Voevodsky’s axiomatization of heigher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hibert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in Categorical logic opens new possibilities for using this method in physics and other natural sciences.



A basic formula for prime numbers

Por • 19 sep, 2012 • Category: Educacion

The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a basic formula for prime numbers, they are identified, and it has been shown that prime numbers, under a special and systematic procedure, are combinations (unions and intersections) of some subsets of natural numbers, with elementary structures. In fact, logical essence of obtained formula for prime numbers is quite similar to formula 2n for even numbers and formula 2n -1 for odd numbers. Usually the prime numbers appear complex and irregular, here they have been described and more clarified, and finally specified examples for obtained formula are presented.



Structural connections between a forcing class and its modal logic

Por • 5 ago, 2012 • Category: Crítica

The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, interpreting necessity as «in all forcing extensions» and possibility as «in some forcing extension». In this modal language one may easily express sweeping general forcing principles, such as the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC (Stavi-V\»a\»an\»anen, Hamkins). Every definable forcing class similarly gives rise to the corresponding forcing modalities, for which one considers extensions only by forcing notions in that class. In previous work, we proved that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2.



Two Paths to Infinite Thought: Alain Badiou and Jacques Derrida on the Question of the Whole

Por • 25 jul, 2012 • Category: Filosofía

This essay defends an idea that is no longer fashionable: that there is a whole. The motivation for a defense of this notion has nothing to do with intellectual conservatism or a penchant for Hegel. Rather, what we hope to establish is a second path into what Alain Badiou has called the ‘Cantorian Revolution’. In order to open this path we undertake a three-fold task. First, we deconstruct Badiou’s onto-logical project by isolating the suppressed significance of Ernst Zermelo. This point allows us to recover a Cantorian possibility for addressing the infinite as an inconsistent whole. Second, we turn to work by the logician Graham Priest in order to remove the absurdity of discussing true contradictions. Finally, we return to Jacques Derrida’s early work on Husserl in order to chart a phenomenological path to an affirmation of an inconsistent whole. We close, then, with the implications for contemporary philosophy.



Constructivist and Structuralist Foundations: Bishop’s and Lawvere’s Theories of Sets

Por • 31 ene, 2012 • Category: Educacion

Bishop’s informal set theory is briefly discussed and compared to Lawvere’s Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive type theory of Martin-L\»of. The theory, CETCS, provides a structuralist foundation for constructive mathematics in the style of Bishop.



Formalizing set theory in weak logics, searching for the weakest logic with Gödel’s incompleteness property

Por • 16 nov, 2011 • Category: Opinion

We show that first-order logic can be translated into a very simple and weak logic, and thus set theory can be formalized in this weak logic. This weak logical system is equivalent to the equational theory of Boolean algebras with three commuting complemented closure operators, i.e., that of diagonal-free 3-dimensional cylindric algebras (Df_3’s). Equivalently, set theory can be formulated in propositional logic with 3 commuting S5 modalities (i.e., in the multi-modal logic [S5,S5,S5]). There are many consequences, e.g., free finitely generated Df_3’s are not atomic and [S5,S5,S5] has G\»odel’s incompleteness property. The results reported here are strong improvements of the main result of the book: Tarski, A. and Givant, S. R., Formalizing Set Theory without variables, AMS, 1987.



Teoría de conjuntos y ontología

Por • 10 oct, 2011 • Category: Leyes

Preguntarse qué clase de entidad es una teoría científica es plantearse una pregunta ontológica. Su respuesta consistirá de categorías ontológicas que subsumirán a las teorías. Si tales son las de la teoría de conjuntos, esto significa que las teorías científicas son entidades conjuntistas, al menos en parte. En este trabajo construyo tres modos posibles de elucidar la pretensión de la teoría de conjuntos como ontología para las teorías científicas. La primera es una lectura de la teoría de conjuntos como ontología formal en el sentido de Husserl. En la segunda, en base a Tugendhat (2003), el uso de la teoría de conjuntos para la reconstrucción de teorías significa que ésta es una semántica formal. La tercera es la presentación de una lectura ajena al estructuralismo, pero que, sin embargo, considera a la teoría de conjuntos como ontología. Lo que hago aquí es utilizar la propuesta de Badiou para pensar, desde una vía alternativa a la tesis ontosemántica, la actividad de reconstruir teorías utilizando a la teoría de conjuntos



Complete Totalities

Por • 22 jul, 2011 • Category: Filosofía

The cumulative hierarchy conception of set, which is based on a metaphor of elements of sets being prior to their collection, is generally considered to be a good way to create a set conception that seems safe from contradictions. This imposes two restrictions on sets. One is a «limitation of size,» and the other is the rejection of non well-founded sets. Quine’s NF system of axioms, does not have any of the two restrictions, but it has a formal restriction on allowed formulas in its comprehension axiom schema, which reflects a similar notion of elements being prior to sets. The suggestion made here is that a possible reason for set antinomies is the tension between our perception of sets and their elements as lying on a different conceptual planes, and our wish to be able to refer to mathematical objects without contemplating their relation with other objects. A new approach to sets as totalities is presented which is based on a notion of «concurrent aggregation,» which instead of avoiding «viscous circles,» acknowledges the inherent circularities of some predicates, and provides a way to learn from their natural emergence.



EL PROBLEMA DEL CONTINUO DESPUÉS DE COHEN (1964-2004)

Por • 19 jun, 2011 • Category: Crítica

En este trabajo se expone en que consiste el nuevo axioma llamado Martin Máximo Acotado (BMM)1, el cual es un axioma que puede considerarse «natural» en cierto sentido y que junto con la teoría ZFE decide el problema del continuo de Cantor. El llamado Axioma de Martin (AM) es un conocido enunciado relacionado con la topología, la combinatoria infinita y el forcing, planteado por Donald Martin en 1970. En 1988 Foreman, Magidor y Shelah, formularon una versión fuerte maximal de AM y lo llamaron Martin Máximo (MM). También demostraron la consistencia de MM relativa a la existencia de un cardinal supercompacto. BMM es una modificación acotada de MM que resulta más débil y que decide el problema del continuo, en el sentido de que el cardinal del continuo es aleph 2