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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.
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: 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.
Received Day Month Year Revised Day Month Year Communicated by Managing Editor
, 707
"... 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 ..."
Abstract
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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.
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"... incompleteness theorem, and an inherent limit on the predictability of evolution ..."
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incompleteness theorem, and an inherent limit on the predictability of evolution
On Ontology and Realism in Mathematics * Outline
"... The paper is concerned with the way in which “ontology ” and “realism ” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possib ..."
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The paper is concerned with the way in which “ontology ” and “realism ” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possible coherent positions, their strengths and weaknesses. I shall also discuss related but different aspects of these problems. The terms in the title are the common thread that connects the various sections. The discussed topics range widely. Certain themes repeat in different sections, yet, for most part the sections (and sometimes the subsections) can be read separately Section 1 however states the basic position that informs the whole paper and, as such, should be read (it is not too long). The required technical knowledge varies, depending on the matter discussed. Aiming at a broader audience I have tried to keep the technical requirements at a minimum, or to supply a short overview. Material, which is elementary for some readers, may be far from elementary for others (my apologies to both). I also tried to supply information that may be of interest to everyone interested in these subjects.
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
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
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