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برهان غودل ومسألة اللاتمامية رأب الصدع في الرياضيات
في عام 1931 نشرت ورقة بحثية قصيرة في الدورية العلمية الألمانية تحت عنوان : \"حول قضايا صورية لا يمكن البت فيها من مبادئ الرياضيات والأنساق المتعلقة بها\". لمؤلف رياضياتي شاب في الخامسة والعشرين من عمره في جامعة فيينا يدعى كورت غودل Kurt Gödel، ومنذ العام 1938 صار عضواً دائماً بمعهد الدراسات المتقدمة في برنستون-الولايات المتحدة الأمريكية. كما تعد هذه الورقة بمثابة حجر الزاوية في تاريخ المنطق والرياضيات؛ وفي العام 1952 منحت جامعة هارفارد هذا الرجل الدرجة الفخرية، وعدت أعماله كواحدة من الدراسات المتقدمة والهامة في منطق العصر الحديث، تجدر الإشارة إلى أن ورقة غودل شنت هجوماً عنيفاً على مسألة مركزية في أسس الرياضيات، لذا بات من الأفضل أن نعطي ملخصاً أولياً عن هذه المسألة.
Book
Gödel's Theorem
Among the many expositions of Gödel's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzén gives careful, non-technical explanations both of what those theorems say and, more importantly, what they do not. No other book aims, as his does, to address in detail the misunderstandings and abuses of the incompleteness theorems that are so rife in popular discussions of their significance. As an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it is a valuable addition to the literature. John W. Dawson, author of Logical Dilemmas: The Life and Work of Kurt Gödel
eBook
Axiomatization of Crisp Gödel Modal Logic
In this paper we consider the modal logic with both □ and ◇ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Gödel algebra [0,1]G. We provide an axiomatic system extending the one from Caicedo and Rodriguez (J Logic Comput 25(1):37-55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given too. We also prove that in the studied logic it is not possible to get ◇ as an abbreviation of □, nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature.
Journal Article
Kurt Gödel and the foundations of mathematics : horizons of truth
\"This volume commemorates the life, work, and foundational views of Kurt Gödel (1906-1978), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances, and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology, and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy, and other disciplines for future generations of researchers\"-- Provided by publisher.
BOOK
A Note on Gödel-Dummet Logic LC
Let \\(A_0,A_1,...,A_n\\) be (possibly) distintict wffs, \\(n\\) being an odd number equal to or greater than 1. Intuitionistic Propositional Logic IPC plus the axiom \\((A_0 A_1) ... (A_n-1 A_n) (A_n A_0)\\) is equivalent to Gödel-Dummett logic LC. However, if \\(n\\) is an even number equal to or greater than 2, IPC plus the said axiom is a sublogic of LC.
Journal Article
Entanglement, Microcausality and Gödel’s Theorem
One of the key points of Pauli’s proof of the spin-statistics theorem is the principle of microcausality , which essentially states “that all physical quantities at finite distance exterior to the light cone ( for | x o ′ − x o ″ | < | x ′ − x ″ | ) are commutable ”. Indeed, Pauli was aware that if it were not valid then neither was his version of the spin-statistics theorem. In this presentation, we explore the relationship between entanglement and microcausality and point out that in the case of spin-singlet states, microcausality does not apply. Consequenly, we revise the spin statistics theorem to incorporate entanglement and to suggest some refinements to the axiomatic structure of quantum mechanics. Ironically, singlet states are SL (2, C ) invariant as is the Minkowski metric of special relativity, although the singlet state is often used to convey “spooky action at a distance” as if it were in violation of special relativity, which it is not. We also pose the question whether the paradoxes associated with “entanglement” can be understood as a special case of Gödel’s theorem.
Journal Article
A Logic for Dually Hemimorphic Semi-Heyting Algebras and its Axiomatic Extensions
The variety \\(\\mathbb{DHMSH}\\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \\(\\mathbb{DHMSH}\\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called \"Dually hemimorphic semi-Heyting logic\" (\\(\\mathcal{DHMSH}\\), for short), as an expansion of semi-intuitionistic logic \\(\\mathcal{SI}\\) (also called \\(\\mathcal{SH}\\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \\(\\mathbb{DHMSH}\\). It is deduced that the logic \\(\\mathcal{DHMSH}\\) is algebraizable in the sense of Blok and Pigozzi, with the variety \\(\\mathbb{DHMSH}\\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \\(\\mathcal{DHMSH}\\) is dually isomorphic to the lattice of subvarieties of \\(\\mathbb{DHMSH}\\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \\(\\mathcal{DHMSH}\\) in which the \"Deduction Theorem\" holds. Thirdly, we present several new logics, extending the logic \\(\\mathcal{DHMSH}\\), corresponding to several important subvarieties of the variety \\(\\mathbb{DHMSH}\\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.
Journal Article
Kripke's Gödel case
Kripke has taken the Gödel case as a counterexample for reference descriptivism. Machery et al. question the validity of Kripke's case and had conducted empirical studies to show its inadequacy. Experimental data suggest intuitions on this matter vary both across and within cultures. However, there is a descriptive ambiguity, we argue, in Kripke's Gödel case, for people associate different types of descriptions with proper names, such as the description of brute facts and the description of social facts. We argue in this paper with experimental data that the descriptive ambiguity exists and affects the actual ratio of Kripkeans in reference. This result flaws Machery et al.'s interpretation on empirical research, but does not challenge their claim on cross-cultural divergence. In fact, there are more East Asian descriptivists than Machery et al. expected.
Kripke toma el caso Godel como un contraejemplo a las teorias descriptivistas de la referencia. Machery et al. cuestionan la validez del caso presentado por Kripke, y han llevado a cabo estudios empiricos para mostrar su inadecuacion. Los resultados experimentales sugieren que las intuiciones sobre estas cuestiones varian tanto entre culturas como dentro de ellas. Sin embargo, sostenemos que existe una ambiguedad descriptiva en el caso Godel de Kripke, ya que los sujetos asocian distintos tipos de descripciones a los nombres propios, por ejemplo, descripciones de hechos brutos y descripciones de hechos sociales. En este articulo defendemos con datos experimentales que dicha ambiguedad descriptiva existe y afecta a la proporcion de kripkeanos sobre la referencia. Este resultado socava la interpretacion de los resultados empiricos realizada por Machery et al., aunque no cuestiona su tesis sobre divergencias entre culturas. De hecho, hay mas descriptivistas entre asiaticos orientales de lo esperado por Machery et al.
Journal Article