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 true-or-false 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 adjusting-to-the-environment 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 real-life 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 real-world fuzziness. After these example, we provide an explanation of this fuzziness – in terms of cognizability of the world. 1

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.

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.

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 stress-energy 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 no-progress report 2 1.

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 stress-energy 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.

incompleteness theorem, and an inherent limit on the predictability of evolution