We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey, Griffiths and LaForte.
We show that van Lambalgen's Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen's Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen's Theorem holds with respect to recursive randomness.
We study the connection between Schnorr triviality and genericity. We show that while no 2-generic is Turing equivalent to a Schnorr trivial and no 1-generic is tt-equivalent to a Schnorr trivial, there is a 1-generic that is Turing equivalent to a Schnorr trivial. However, every such 1-generic must be high. As a corollary, we prove that not all K-trivials are Schnorr trivial. We also use these techniques to extend a previous result and show that the bases of cones of Schnorr (...) trivial Turing degrees are precisely those whose jumps are at least 0". (shrink)
A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null ${\Sigma^0_1}$ (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if ${{\bf h'} \geq_T {\bf 0''}}$ , then h (...) contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees. (shrink)
We investigate the relationship between lowness for Schnorr randomness and Schnorr triviality. We show that a real is low for Schnorr randomness if and only if it is Schnorr trivial and hyperimmune free.
The weakly random reals contain not only the Schnorr random reals as a subclass but also the weakly 1-generic reals and therefore the n -generic reals for every n . While the class of Schnorr random reals does not overlap with any of these classes of generic reals, their degrees may. In this paper, we describe the extent to which this is possible for the Turing, weak truth-table, and truth-table degrees and then extend our analysis to the Schnorr random and (...) hyperimmune reals. (shrink)
We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees (...) of categoricity is $\Pi_{1}^{1}$-complete. (shrink)
We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of back-and-forth relations), jump hierarchies, and computing descending sequences in linear orders.
En el ámbito de la lógica matemática existe un problema sobre la relación lógica entre dos versiones débiles del Axioma de elección (AE) que no se ha podido resolver desde el año 2000 (aproximadamente). Tales versiones están relacionadas con ultrafiltros no principales y con Propiedades Ramsey (Bernstein, Polarizada, Subretículo, Ramsey, Ordinales flotantes, etc). La primera versión débil del AE es la siguiente (A): “Existen ultrafiltros no principales sobre el conjunto de los números naturales (ℕ)”. Y la segunda versión débil del (...) AE es la siguiente (B): “Existen ultraflitros sobre ℕ”. Se sabe que A implica B, pero se desconoce si B implica A. Di Prisco y Henle conjeturan en los artículos ([1], [2]) que esto no ocurre, es decir, conjeturan que B no implica A, en otras palabras, conjeturan que A es más fuerte estrictamente que B, que A es independiente de B, pero esto no se ha podido demostrar todavía aunque se ha intentado hacer desde hace aproximadamente 21 años. Una descripción detallada de este problema abierto puede encontrarse en esta ponencia (dictada en el marco del Día Mundial de la Lógica 14-01-2022) y en el artículo [3]. [1] C. Di Prisco y H. Henle. “Doughnuts, Floating Ordinals, Square Brackets, and Ultraflitters”. Journal of Symbolic Logic 65 (2000) 462-473. [2] C. Di Prisco y H. Henle. “Partitions of the reals and choice”. En “Models, algebras and proofs”. X. Caicedo y C.M. Montenegro. Eds. Lecture Notes in Pure and Appl. Math, 203, Marcel Dekker, 1999. [3] F. Galindo. “Tópicos de ultrafiltros”. Divulgaciones Matemáticas. Vol. 21, No 1-2, 2020. (shrink)
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/o de fundamentos de la matemática. El destacado matemático Joan Bagaria afirma lo siguiente sobre el método de forcing en su artículo "Paul Cohen y la técnica del forcing" (Gaceta de la Real Sociedad Matemática Española, Vol. 2, Nº 3, 1999, págs 543-553) : "Aunque (...) Cohen recibió la medalla Fields por su demostración de la independencia de la hipótesis del continuo y del axioma de elección, su contribución va mucho más allá de estos problemas. Su nuevo método, el forcing, no sólo ha permitido resolver un sinfín de problemas importantes en prácticamente todas las áreas de las matemáticas, si no que ha cambiado para siempre nuestra concepción de la matemática como ciencia". El objetivo principal del siguiente trabajo es estudiar el método de forcing describiendo algunas de sus aplicaciones (forcing de Cohen, forcing aleatorio, forcing de Mathias, forcing de Sacks, forcing de Silver, etc.), y ofreciendo una aproximación a sus fundamentos metamatemáticos. Se aspira que estas notas sirvan de apoyo para aprender dicho método. (shrink)
Ultrafilters are very important mathematical objects in mathematical research [6, 22, 23]. There are a wide variety of classical theorems in various branches of mathematics where ultrafilters are applied in their proof, and other classical theorems that deal directly with ultrafilters. The objective of this article is to contribute (in a divulgative way) to ultrafilter research by describing the demonstrations of some such theorems related (uniquely or in combination) to topology, Measure Theory, algebra, combinatorial infinite, set theory and first-order logic, (...) also formulating some updated open problems of set theory that refer to non-principal ultrafilters on N, the Mathias’s model and the Solovay’s model. -/- Resumen: Los ultrafiltros son objetos matemáticos muy importantes en la investigación matemática [6, 22, 23]. Existen una gran variedad de teoremas clásicos en diversas ramas de la matemática donde se aplican ultrafiltros en su demostración, y otros teoremas clásicos que tratan directamente sobre ultrafiltros. El objetivo de este artículo es contribuir (de una manera divulgativa) con la investigación sobre ultrafiltros describiendo las demostraciones de algunos de tales teoremas relacionados (de manera única o combinada) con topología, teoría de la medida, álgebra, combinatoria infinita, teoría de conjuntos y lógica de primer orden, formulando además algunos problemas abiertos actuales de la teoría de conjuntos que se refieren a ultrafiltros no principales sobre N, al Modelo de Mathias y al Modelo de Solovay. (shrink)
Definición del derecho natural -- Definición del derecho de gentes -- Diferencia de otras ciencias -- Del derecho público -- De la política -- Necesidad y utilidad del derecho natural y de gentes -- Origen y antiguedad de este derecho -- Formación de esta ciencia -- Progresos y escritores principales. Hugo Grotius -- John Selden -- Thomas Hobbes -- Samuel Pufendorf -- Christian Thomasius -- Johann Gottlieb Heineccius -- Christian Wolff -- Emer de Vattel -- Jean-Jacques Burlamaqui. Fortunato Bartolomeo De (...) Felice -- Otros escritores en com un -- Montesquieu -- Simon-Nicolas-Henri Linguet -- Jean-Jacques Rousseau -- Escritos modernos detestables -- Vicios y defectos de muchos modernos -- Modo para conocer los autores sospechosos -- Autores católicos con que se refutan -- Precaución para establecer las reglas del derecho natural -- Falsa preocupaci on de buscar un primer principio -- Métodos que se proponen -- No son necesarios -- Del método escolástico -- Universidades en que se enseña este derecho. (shrink)
_The purpose_ of this study is the reflection on ways of philosophical legitimation for the "Self-made-man" idea in the worldview of the Renaissance and Reformation. _Theoretical basis._ Historical, comparative, and hermeneutic methods became the basis for this. The study is based on the works of Nicholas of Cusa, G. Pico della Mirandola, N. Machiavelli, M. Montaigne, E. Roterodamus, M. Luther, J. Calvin together with modern researchers of this period. _Originality._ The analysis allows us to come to the conclusion that casts (...) doubt on the still widespread belief according to which the emergence of the "Self-made-man" idea is localized by the process of forming the American cultural code and the ideological heritage of Benjamin Franklin. It is highlighted that the formation of this idea is the result of a long process that originated in the ancient world and gains a special impetus in the Renaissance and Reformation. Precisely in the cultural context of the latter, the ancient intention to recognize the individual’s right to self-determination and self-government, which in the depths of Christian theology acquires only a potentially universal character, becomes not only acceptable but also, in the context of Protestantism worldview, the only admissible, in fact, individual’s obligatory life guidelines. _Conclusions._ Humanistic and reformation thoughts quite naturally led to further ideological legitimation of the person’s idea of who is creating oneself. This legitimation was during the complex interaction of numerous factors of culture in the Late Middle Ages, as well as ideas and intentions inherited from Antiquity. Key among them was the gradual formation of a new social order, in essence, indifferent to paternalistic rudiments, together with the ethics of Protestantism corresponding to it. The latter does not only legalize but, de facto, sacralizes the individual’s reorientation from hopes for the synergy of God’s grace and own free will in personal salvation, toward the self-reliance and personal efforts, awareness of personal responsibility for the own fate as key principles of the "Self-made-man" concept. (shrink)
Abstract Internet communication technology has been said to affect our sense of self by altering the way we construct “personal identity,” understood as identificatory valuative narratives about the self; in addition, some authors have warned that internet communication creates special conditions for moral agency that might gradually change our moral intuitions. Both of these effects are attributed to the fact that internet communication is “disembodied.” Our aim in this paper is to establish a link between this complex of claims and (...) past and ongoing research in phenomenology, empirical psychology and cognitive science, in order to formulate an empirical hypothesis that can assist development and evaluation of recent technology for embodied telecommunication. We first suggest that for the purposes of interdisciplinary exchange, personal identity is formally best represented by a selection function that (for temporal intervals of variable length) “bundles” capacity ascriptions into identificatory narratives. Based on this model, we discuss which cultural changes engendered by the internet affect the construction of personal identity in ways that diminish our ethical sensitivies. In a second step, working from phenomenological claims by Martin Buber, we argue that disembodied communication severs two modes of cognitive function, preconceptual and conceptual, which tie together moral motivation, self-experience, and identity construction. We translate Buber’s claims into the theoretical idiom of the “theory of cognitive orientation,” a psychological theory of motivation that links up with recent research in embodied cognition. In a third step, we investigate whether the embodiment of the internet with communication robots (e.g., telenoids) holds out the prospect of reverting this structural change at least partially. We conclude by formulating an empirical hypothesis (for researchers in cognitive science) that has direct import, we submit, on the question whether embodied telecommunication promises a new form of ethically sensitive self-constituting encounter. Content Type Journal Article Category Special Issue Pages 1-23 DOI 10.1007/s13347-012-0064-9 Authors Johanna Seibt, Department for Philosophy and the History of Ideas, Aarhus University, Aarhus, Denmark Marco Nørskov, Department for Philosophy and the History of Ideas, Aarhus University, Aarhus, Denmark Journal Philosophy & Technology Online ISSN 2210-5441 Print ISSN 2210-5433. (shrink)
Las dificultades teóricas que obstaculizan la idea de una ética constitucional -- ¿constitucionalismo global o constitucionalismo cosmopolita? -- El constitucionalismo, entre el cosmopolitismo, el universalismo y el localismo del estado nación -- El estado constitucional de derecho (o Estado constitucional de derecho, social y ambiental) y los derechos sociales -- El impacto del consenso de Washington en el derecho social colombiano, y la respuesta social desde la movilización y desde los derechos -- Los derechos, un espacio de lucha y emancipación (...) -- La autocrítica a modo de conclusión. (shrink)
Intuición y hermenéutica -- El ser de sí mismo en la identidad narrativa de Paul Ricoeur -- Fenomenología como hermenéutica de la existencia -- Filosofía de la religión -- Los fines de la educación -- Tema formadores y medios -- Las experiencias de aprendizaje artístico de los estudiantes.
A highly idiosyncratic search for universal laws governing the development of a feeling of well-being and creative involvement in life. Numerous, somewhat mystical, experiences are described with enthusiasm and candor, but, perhaps, uncertain significance for those less inclined toward mysticism.—N. Y.
Spivak translates Derrida’s “il n’y a pas de hors-texte” as “there is nothing outside the text.” By considering how the aphorism works within his study of Rousseau on sexual and textual supplements, and by reviewing related expressions in French, a mistranslation is revealed. This is not a simple error, however. The distortion is generated by Derrida’s own broader context. We must not only distinguish signification from reference but also place the aphorism within Derrida’s allusion, in the first part of Of (...) Grammatology, to an all-embracing arche-writing. The paper ends in thus opening out the discussion of a textual “inside” or “outside.”. (shrink)
Sudoku puzzles, which are popular worldwide, require individuals to infer the missing digits in a 9 9 array according to the general rule that every digit from 1 to 9 must occur once in each row, in each column, and in each of the 3-by-3 boxes in the array. We present a theory of how individuals solve these puzzles. It postulates that they rely solely on pure deductions, and that they spontaneously acquire various deductive tactics, which differ in their difficulty (...) depending on their “relational complexity”, i.e., the number of constraints on which they depend. A major strategic shift is necessary to acquire tactics for more difficult puzzles: solvers have to keep track of possible digits in a cell. We report three experiments corroborating this theory. We also discuss their implications for theories of reasoning that downplay the role of deduction in everyday reasoning. (shrink)
« En vertu de quoi la catégorie de l’Utopique possède donc à côté de son sens habituel et justement dépréciatif, cet autre sens qui, loin d’être nécessairement abstrait et détourné du monde, est au contraire centralement préoccupé du monde : celui du dépassement de la marche naturelle des événements »« Il n’y a guère, même parmi les économistes bourgeois, un savant..
Si on cherche une définition minimale de la phénoménologie, on trouvera l'idée de philosophie descriptive. En ce sens, on peut imputer quelque chose de phénoménologique á de nombreuses philosophies contemporaines, audelà des limites de la phénoménologie au sens étroit et historique du terme. Néanmoins, il faut alors prendre en compte la question de savoir ce qui est à décrire. La réponse phénoménologique est: ce qui est vu. Qu'est-ce qui est donc vu ? Cette question est inséparable de celle de la (...) grammaire du voir. L'auteur soutient que c'est ce qui doit être retenu de la critique wittgensteinienne de la phénoménologie. Mais, si ce qui doit être dit est le vu en tant qu'il est dit, il n'en reste pas moins qu'il doit être dit en tant que vu. Que faire de cette dimension qui est celle de l'« être-vu » ? En ce point, la phénoménologie est inévitable, non plus comme discipline, mais au moins comme problème. If one requires a minimal definition for phenomenology, one may settle for the idea of a descriptive philosophy. In that sense, something phenomenological can be ascribed to many contemporary species of philosophiy, beyond the limits of « phenomenology » taken in a narrow, historical sense. However, the question arises : what is to be described? Phenomenology's answer is: what is seen. What, then, is seen ? This question is inseparable from the question of the grammar of seeing, as the author holds to be drawn from the wittgensteinian criticism of phenomenology. But if what is to be said is what is seen, as said, yet it remains that it is to be said as seen. What can be done with this « being-seen-ness » ? Here, phenomenology is unavoidable, not as a discipline anymore, but at least as a problem. (shrink)
