Ricardo Da Silva

Church's Undecidability Theorem is one of the meta-theoretical results of the mid-third decade of the last century, which along with other limiting theorems such as those of Gödel and Tarski have generated endless reflections and analyzes, both within the framework of the formal sciences, that is, mathematics, logic and theoretical computation, as well as outside them, especially the philosophy of mathematics, philosophy of logic and philosophy of mind. We propose, as a general purpose of this article, to formulate Church's Undecidability Theorem and present the main ideas of its demonstration. In order to carry out the first objective, we need to introduce and explain the notions of recursive function and numbering used by Gödel, which will allow to formally and rigorously enunciate Church's Theorem. After we enunciate Church's Theorem of Unspeakability in a formal and rigorous manner, we will present the main ideas of the proof of Church's Undecidability Theorem for First Order Logic, which uses Robinson's axiomatic system for arithmetic and four facts about himself: In Robinson's system for arithmetic recursive functions are representable Robinson's system is undecidable, The number of axioms proper to the Robinson system is finite and The logical calculation of the Robinson system is equal to the calculation of the first-order logic.

The present paper has three objectives: Presenting an actualization of a proof of the decidability of monadic predicates logic in the contemporary model theory context; Show examples of decidable and undecidable fragments inside First order logic, offering an original proof of the following theorem: Any formula of First of order logic is decidable if its prenex normal form is in the following form: ∀x1,…,∀xn∃y1,…,∃ymφ; Presenting a theorem that characterizes the validity of First order logic by the tautologicity of Propositional logic, said result is interesting since immediately arises the doubt of how to conciliate said characterization with Alonzo Church’s Undecidability Theorem for First Order Logic.

Os casos de fraqueza da vontade (incontinência) são tidos como paradoxais. Há, pelo menos, uma tensão entre tais casos e a lei psicológica que relaciona avaliações e volições: se julgamos que fazer x é melhor do que fazer y, então queremos fazer x; porém, por vezes, julgamos que fazer x é melhor do que fazer y e, ainda assim, queremos fazer y em vez de x. Como noutras propostas, defendo que a incompatibilidade é aparente. Porém, não rejeito que haja verdadeiros casos de fraqueza da vontade. Proponho que, em tais casos, há uma cisão entre a força psicológica e o conteúdo axiológico das avaliações: o que o agente representa como mais valioso não é o que se encontra mais representado como valioso no agente e, portanto, aquilo que tem eficácia causal. Proponho, ainda, que o sistema volitivo desenvolveu um mecanismo de correção de tais anomalias, pelo qual o agente tem a capacidade de determinar volições de acordo com a avaliação do melhor.

According to the principle of special relativity, the laws of nature are the same for systems at rest and for systems in uniform and rectilinear motion: for the formulation of the laws of nature, the body C and the body C´, which, relative to the former, is in uniform and rectilinear motion, are equivalent. Galileo, in the Dialogue on the Two Greatest Systems of the World, developed the following argument in favor of such a principle: two bodies C and C´, which are in uniform and rectilinear motion in relation to a third body C´´, with the same direction and speed, are at rest in relation to one another; therefore, what is applicable to rest, is applicable to uniform and rectilinear motion. Einstein, in On the Influence of Gravity in the Propagation of Light, generalizes the principle of relativity: all reference bodies C, C´, etc., are equivalent for the formulation of the laws of nature, whatever their state of motion is. The argument is analogous to that of Galileo: two bodies in free fall are at rest in relation to each other; therefore, what is applicable to rest, is applicable to constant acceleration. I argue that those arguments are invalid: to say that rest is shared motion does not mean that rest is, simply, movement. Therefore, one can only conclude that the laws of nature are the same for systems at rest and for systems with shared motion.

The turbulent times we live in, product of the COVID-19 pandemic, creating an ideal breeding ground for the proliferation of myths and conspiracy theories. These myths and theories are often popular on social media, defended by irresponsible politicians and published by the media in search of ratings or likes. But none of these meet the theoretical virtues that a good scientific theory should enjoy, in particular, none of these conspiracy theories about the disease caused by the SARS CoV-2 virus has the attribute of being simple. Our purpose in this article is to illuminate how it is possible that simplicity, within the framework of theoretical virtues, can function as an epistemic criterion to reveal the irrational and, consequently, unscientific nature of myths and conspiracy theories.

Hacia finales del siglo XIX se llevó a cabo una gran revolución conceptual y metodológica en la matemática. En tal revolución se empezaron a emplear conceptos, métodos y técnicas que dejaban de lado la antigua forma de hacer matemática, propia del siglo XVIII y principios del siglo XIX, y a su vez proponían un Hacer abstracto, es decir, una forma abstracta de ocuparse del ente matemático. Pero no sólo se trataba de un cambio metodológico, sino que la pregunta por los fundamentos se vuelve cada vez más importante y trae consigo interrogantes de carácter filosófico, como es el caso de la inquietud por la naturaleza del objeto matemático (la interrogante ontológica), y la posibilidad de conocimiento de dicho objeto (la interrogante epistemológica). Nuestro interés en este artículo es mostrar cómo la filosofía que respalda las investigaciones matemáticas de Cantor trata de dar respuestas a las interrogantes ontológicas y epistemológicas. Para ello hemos tratado de ofrecer un contexto histórico-conceptual que gira en torno a la pregunta por los fundamentos, y dentro de dicho contexto hemos señalado como se presenta el Platonismo absoluto de Cantor An approach to Cantor´s absolute PlatonismA great conceptual and methodological revolution in mathematics was carried out by the end of nineteen century. In that revolution people began to use concepts, methods and techniques which set aside the old way of doing mathematics, typical of the eighteenth and early nineteenth century, and in turn they proposed an Abstract Make, i.e., an abstract form of dealing with the mathematical entity. But it was not only a methodological change, but the question of the foundations is becoming increasingly important which arises more philosophical questions, such as the concern about the nature of the mathematical object -the ontological question- and the possibility of knowledge of this object -the epistemological question. Our interest in this article is to show how the philosophy behind Cantor's mathematical research is intended to answer the ontological and epistemological questions. For it, this paper tries to provide a conceptual and historical context, which is revolving around the question of the foundations, and within this context, it is noted as the Absolute Platonism of Cantor

This paper aims to provide an adequate philosophical groundwork to normativity. More specifîcally, the author argues in favor of its axiological nature: from a philosophical point of view, normativity only becomes intelligible if values are taken into account. Therefore values arise as a necessary condition of normativity. Although not sufficient, values are the decisive condition in distinguishing norms from other relations of the same type. The thought of axiologists like Max Scheler and Nicolai Hartmann is discussed even though their theories do not fully explain how norm and value interrelate.

The following note has on purpose to introduce interested students to logicism. Our objective is not to show any new interpretation or thesis about logicism or its rebirth between the 60s and 80s of the last century. What we will do is systematically show the evolution of logicism from Frege to Russell-Whitehead, with greater emphasis on this latest development, and approach some problems that arise within that movement, for example: The logical paradoxes and the principle of intuitive comprehension, the impredicative definitions and the semantic paradoxes, the Ramified Hierarchy and real numbers, the axiom of reducibility and the impossibility of reducing mathematics to logic, etc.