Interpretability formalized
... and for different purposes. A famous and well known example is an interpretation of hyperbolic geometry in Euclidean geometry (e.g., the Beltrami-Klein model, see, for example, [Gre96]) to show the relative consistency of non-Euclidean geometry. Another example, no less famous, is Gödel’s interpret ...
... and for different purposes. A famous and well known example is an interpretation of hyperbolic geometry in Euclidean geometry (e.g., the Beltrami-Klein model, see, for example, [Gre96]) to show the relative consistency of non-Euclidean geometry. Another example, no less famous, is Gödel’s interpret ...
Predicate Logic
... Two statements S and T involving predicates and quantifiers are logically equivalent if and only if they have the same truth value regardless of the interpretation, i.e. regardless of the meaning that is attributed to each propositional function, the domain of discourse. ...
... Two statements S and T involving predicates and quantifiers are logically equivalent if and only if they have the same truth value regardless of the interpretation, i.e. regardless of the meaning that is attributed to each propositional function, the domain of discourse. ...
Untitled
... (3) The chapter on the construction of the natural numbers, integers and rational numbers from the Peano Postulates was removed entirely. That material was originally included to provide the needed background about the number systems, particularly for the discussion of the cardinality of sets in Cha ...
... (3) The chapter on the construction of the natural numbers, integers and rational numbers from the Peano Postulates was removed entirely. That material was originally included to provide the needed background about the number systems, particularly for the discussion of the cardinality of sets in Cha ...
A Course in Modal Logic - Sun Yat
... concepts, whereas these differences are actually supported by different intuitive semantics (even philosophical background). The reader who is interested in this can refer to the related literature. (Ⅱ) The cardinality of At is finite or countable infinite, but, in fact, most of results given in thi ...
... concepts, whereas these differences are actually supported by different intuitive semantics (even philosophical background). The reader who is interested in this can refer to the related literature. (Ⅱ) The cardinality of At is finite or countable infinite, but, in fact, most of results given in thi ...
Contents
... of any serious problem solver. There are countless problems that reduce readily to this inequality and even more problems in which the CauchySchwarz inequality is the key idea of the solution. In this unit we will not focus on the theoretical results, since they are too well-known. Yet, seeing the C ...
... of any serious problem solver. There are countless problems that reduce readily to this inequality and even more problems in which the CauchySchwarz inequality is the key idea of the solution. In this unit we will not focus on the theoretical results, since they are too well-known. Yet, seeing the C ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".