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Theory of gravitation theories: a noprogress report
, 707
"... Abstract. Already in the 1970s there where attempts to present a set of ground rules, sometimes referred to as a theory of gravitation theories, which theories of gravity should satisfy in order to be considered viable in principle and, therefore, interesting enough to deserve further investigation. ..."
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Abstract. Already in the 1970s there where attempts to present a set of ground rules, sometimes referred to as a theory of gravitation theories, which theories of gravity should satisfy in order to be considered viable in principle and, therefore, interesting enough to deserve further investigation. From this perspective, an alternative title of the present paper could be “why are we still unable to write a guide on how to propose viable alternatives to general relativity?”. Attempting to answer this question, it is argued here that earlier efforts to turn qualitative statements, such as the Einstein Equivalence Principle, into quantitative ones, such as the metric postulates, stand on rather shaky grounds — probably contrary to popular belief — as they appear to depend strongly on particular representations of the theory. This includes ambiguities in the identification of matter and gravitational fields, dependence of frequently used definitions, such as those of the stressenergy tensor or classical vacuum, on the choice of variables, etc. Various examples are discussed and possible approaches to this problem are pointed out. In the course of this study, several common misconceptions related to the various forms of the Equivalence Principle, the use of conformal frames and equivalence between theories are clarified. Theory of gravitation theories: a noprogress report 2 1.
A MachineAssisted Proof of Gödel’s Incompleteness Theorems for the Theory of Hereditarily Finite Sets
, 2013
"... Abstract. A formalisation of Gödel’s incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows ´Swierczkowski (2003), who gave a detailed proof using hereditarily finite ..."
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Abstract. A formalisation of Gödel’s incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows ´Swierczkowski (2003), who gave a detailed proof using hereditarily finite set theory. The adoption of HF is generally beneficial, but it poses certain technical issues that do not arise for Peano arithmetic. The formalisation itself should be useful to logicians, particularly concerning the second incompleteness theorem, where existing proofs are lacking in detail. §1. Introduction. Gödel’s incompleteness theorems (Feferman, 1986; Gödel, 1931) are undoubtedly the most misunderstood results in mathematics. Franzén (2005) has written an entire book on this phenomenon. One reason is they have attracted the attention of a great many nonmathematicians, but even specialists who should know better have drawn unfounded conclusions. One of the main obstacles to understanding these theorems is the
λCalculus: Then & Now
, 2012
"... Notes derived from the slides presented at the conferences. A brief amount of text has been added for continuity. The author would be happy to hear reactions and suggestions. ..."
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Notes derived from the slides presented at the conferences. A brief amount of text has been added for continuity. The author would be happy to hear reactions and suggestions.
A Mechanised Proof of Gödel’s Incompleteness Theorems using Nominal Isabelle
"... Abstract A Isabelle/HOL formalisation of Gödel’s two incompleteness theorems is presented. Aspects of the development are described in detail, including two separate treatments of variable binding: the nominal package [25] and de Bruijn indices [3]. The work follows ´ Swierczkowski’s a detailed proo ..."
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Abstract A Isabelle/HOL formalisation of Gödel’s two incompleteness theorems is presented. Aspects of the development are described in detail, including two separate treatments of variable binding: the nominal package [25] and de Bruijn indices [3]. The work follows ´ Swierczkowski’s a detailed proof, using hereditarily finite set theory [23]. 1
1 Semiotic Systems, Computers, and the Mind: How Cognition Could Be Computing
"... computationalism, semiotic systems, cognition, syntax, semantics ..."
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computationalism, semiotic systems, cognition, syntax, semantics
Computability and Algorithmic Complexity in Economics
, 2012
"... Rabin's effectivization of the GaleStewart Game [42] remains the model methodological contribution to the field for which Velupillai coined the name Computable Economics more than 20 years ago. Alain Lewis was the first to link Rabin's work with Simon's fertile concept of bounded rat ..."
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Rabin's effectivization of the GaleStewart Game [42] remains the model methodological contribution to the field for which Velupillai coined the name Computable Economics more than 20 years ago. Alain Lewis was the first to link Rabin's work with Simon's fertile concept of bounded rationality and interpret them in terms of Alan Turing's work. Solomonoff (1964), one of the three the other two being Kolmogorov and Chaitinacknowledged pioneers of algorithmic complexity theory, had his starting point in one aspect of what Velupillai [72] came to call the Modern Theory of Induction, an aspect which had its origins in Keynes [23]. Kolmogorov's resurrection of von Mises [80] and the genesis of Kolmogorov complexity via computability theoretic foundations for a frequency theory of probability has given a new lease of life to finance theory [49]. Rabin's classic of computable economics stands in the long and distinguished tradition of game theory that goes back to Zermelo [84], Banach & Mazur [5], Steinhaus [62] and Euwe [14]. This is an outline of the origins and development of the way computability theory and algorithmic complexity theory were incorporated into economic and nance theories. We try to place, in the context of the development of computable economics, some of the classics of the subject as well as those that have,
Gödel's Incompleteness Theorems: A Revolutionary View of the Nature of Mathematical Pursuits
"... The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicia ..."
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The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicians studies mathematics. This paper provides a history of the mathematical developments that laid the foundation for Gödel's work, describes the unique method used by Gödel to prove his famous incompleteness theorem, and discusses the farreaching mathematical implications thereof. 2 I.
Intermediate Degrees are Needed for the World to be Cognizable: Towards a New Justification for Fuzzy Logic Ideas
"... Summary. Most traditional examples of fuzziness come from the analysis of commonsense reasoning. When we reason, we use words from natural language like “young”, “well”. In many practical situations, these words do not have a precise trueorfalse meaning, they are fuzzy. One may therefore be left w ..."
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Summary. Most traditional examples of fuzziness come from the analysis of commonsense reasoning. When we reason, we use words from natural language like “young”, “well”. In many practical situations, these words do not have a precise trueorfalse meaning, they are fuzzy. One may therefore be left with an impression that fuzziness is a subjective characteristic, it is caused by the specific way our brains work. However, the fact that that we are the result of billions of years of successful adjustingtotheenvironment evolution makes us conclude that everything about us humans is not accidental. In particular, the way we reason is not accidental, this way must reflect some reallife phenomena – otherwise, this feature of our reasoning would have been useless and would not have been abandoned long ago. In other words, the fuzziness in our reasoning must have an objective explanation – in fuzziness of the real world. In this paper, we first give examples of objective realworld fuzziness. After these example, we provide an explanation of this fuzziness – in terms of cognizability of the world. 1
I. EVERYTHING IS A MATTER OF DEGREE: ONE OF THE MAIN IDEAS BEHIND FUZZY LOGIC
"... Abstract—One of the main ideas behind fuzzy logic and its applications is that everything is a matter of degree. We are often accustomed to think that every statement about a physical world is true or false – that an object is either a particle or a wave, that a person is either young or not, either ..."
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Abstract—One of the main ideas behind fuzzy logic and its applications is that everything is a matter of degree. We are often accustomed to think that every statement about a physical world is true or false – that an object is either a particle or a wave, that a person is either young or not, either well or ill – but in reality, we sometimes encounter intermediate situations. In this paper, we show that the existence of such intermediate situations can be theoretically explained – by a natural assumption that the real world is cognizable.
Gödel’s Incompleteness Theorems
"... In 1931, when he was only 25 years of age, the great Austrian logician Kurt Gödel (1906– 1978) published an epochmaking paper [16] (for an English translation see [8, pp. 5–38]), in which he proved that an effectively definable consistent mathematical theory which is strong enough to prove Peano’s ..."
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In 1931, when he was only 25 years of age, the great Austrian logician Kurt Gödel (1906– 1978) published an epochmaking paper [16] (for an English translation see [8, pp. 5–38]), in which he proved that an effectively definable consistent mathematical theory which is strong enough to prove Peano’s postulates of elementary arithmetic cannot prove its own