Es conocido que el método de forcing es una de las técnicas de construcción de modelos más importantes de la Teoría de conjuntos en la actualidad, siendo el mismo muy útil para investigar problemas de matemática y de fundamentos de la matemática. El objetivo del siguiente trabajo es estudiar tal método, describir algunas de sus aplicaciones y ofrecer una aproximación a sus fundamentos metamatemáticos. Se aspira que este texto sirva de apoyo para aprender dicho método.

The present paper, taking the cue from the Italian translation of Vergangenheit und Zukunft der Sozialwissenschaften (The Past and Future of Social Sciences), a Schumpeter’s book which was not always well understood in the literature, tries to pose some questions about Schumpeter’s work. Firstly: is it possible, starting from that book, to reconstruct a Schumpeterian theory of scientific development? Subsequently: is Vergangenheit und Zukunft only «a brief outline of what first became the Epochen [der Dogmen– und Methodengeschichte] and finally the History of Economic Analysis», as Elizabeth Boody Schumpeter wrote in the Editor’s Introduction (July 1952) to the History of Economic Analysis (p. XXXII), or should it be read as a complement of Epochen and, possibly, History? Lastly: is it correct to say that Schumpeter’s work had the ambitious objective of developing a ‘comprehensive sociology’ as the eminent Japanese scholar Shionoya did?

We introduce a method to derive theorems from Elementary Number Theory by means of relationships among formal languages. Using σ -algebras, we define what a proof of a number-theoretical statement by Language Theory means. We prove that such a proof can be transformed into a traditional proof in ZFC . Finally, we show some examples of non-trivial number-theoretical theorems that can be proved by formal languages in a natural way. These number-theoretical results concern densely divisible numbers, semi-perimeters of Pythagorean triangles, middle divisors and partitions into consecutive parts.

With the advent and growth of electronic communication, the word “cyberspace” has entered into everyday parlance. But what does this word signify? I begin by sketching an equivalence between physical space and cyberspace, showing that they share the concepts of place, distance, size and route in common. With this mutual framework in place, I go on to examine various theories—substantival, relational, Einsteinian and Kantian—concerning the nature of physical space. We see that, while cyberspace shares some of the properties of physical space isolated by each of these theories, still it cannot be subsumed under any one theory. We also see that cyberspace exhibits several novel properties, projecting it far beyond the scope of any existing theory and setting it apart as an exciting new spatial medium.

This paper provides a complete suite of axioms for a version of set theory that I call Explication. Explication borrows from the two most prominent existing systems of set theory. Explication starts with class variables. After several steps, some of these class variables are identified as sets. The most crucial step is to identify classes which can explicitly say, in a finite statement, which other classes are its members. The reader is cautioned that the suite of axioms presented is a complete treatment of Explication. Statements which may or may not be true in other axiom systems are not significant.

Many undergraduate students of engineering and the exact sciences have difficulty with their mathematics courses due to insufficient proficiency in what we in this paper have termed clear thinking. We believe that this lack of proficiency is one of the primary causes underlying the common difficulties students face, leading to mistakes like the improper use of definitions and the improper phrasing of definitions, claimes and proofs. We further argue that clear thinking is not a skill that is acquired easily and naturally – it must be consciously learned and developed. The paper describes, using concrete examples, how the examination and analysis of classical paradoxes can be a fine tool for developing students’ clear thinking. It also looks closely at the paradoxes themselves, and at the various solutions that have been proposed for them.

Information theory is a mathematical theory of learning with deep connections with topics as diverse as artificial intelligence, statistical physics, and biological evolution. Many primers on the topic paint a broad picture with relatively little mathematical sophistication, while many others develop specific application areas in detail. In contrast, these informal notes aim to outline some elements of the information-theoretic “way of thinking,” by cutting a rapid and interesting path through some of the theory’s foundational concepts and theorems. We take the Kullback-Leibler divergence as our foundational concept, and then proceed to develop the entropy and mutual information. We discuss some of the main foundational results, including the Chernoff bounds as a characterization of the divergence; Gibbs’ Theorem; and the Data Processing Inequality.

One of the most puzzling aspects about the functioning of the floating exchange rate regime of the 1980s has been that huge swings in exchange rate have had only muted effects on anything real. To understand this phenomenon, we study the relationship between communism and value neutrality and monetary neutrality. We find that the symmetry of communism is bound to lead to value neutrality. In the case of value neutrality, the economic man will certainly accept monetary neutrality. If money is neutral in the long run then even if purchasing power parity (PPP) is not valid in the short-run it will valid over the long run. However, without considering the time factor, communism is a kind of symmetry that is almost impossible to achieve. While considering the time factor, the symmetry of communism can be achieved in theory!

Mereology is the theory of wholes and parts. The first formal mereology was developed by Husserl in his third Logical Investigation at the beginning of the twentieth century. In 1916 Stanisław Lesnie ´ wski gave the first axiomatization of a classical extensional formal mereology. That same year, Alfred North Whitehead also gave a sketch of a mereology in “La théorie relationniste de l’espace”. It was developed in the perspective of a theory of space in which the concept of point is no longer considered as primitive, but is built in terms of the relations between objects. This project was then taken up and amplified in the wider perspective of the method of extensive abstraction presented in An Enquiry Concerning the Principles of Natural Knowledge and The Concept of Nature. Afterwards, Whitehead added to what was first a theory of the part-whole relation some definitions of topological notions such as junction.

En el prólogo a la segunda edición de la Crítica de la razón pura, ¿qué ha querido decir Kant con una “revolución del modo de pensar” que se debe aplicar a la meta-física para que tome el camino de la ciencia? Cabe distinguir dos significados: uno de carácter metodológico, que asume que el conocimiento rige a los objetos y ten-dría alcance. Aplicada a la metafísica, supone una transformación más profunda que permite hablar, en segundo lugar, de un significado metafísico específico, que implica que el conocimiento constituye aquello mismo a lo que se llama el objeto de las representaciones.