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Some Facts About Kurt Godel
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A Note on Quine's Principles of Quantification
In his Mathematical logic Quine characterizes the theorems of quantification theory by six principles:
38679232
Short Definitions of the Ordinals
In this paper, a simple inductive characterization of the ordinal numbers is stated and developed. The characterization forms the basis for a set of axioms for ordinal theory and also for several short explicit definitions of the ordinals. The axioms are shown to be sufficient for ordinal theory, and, subject to suitable existence assumptions, each of the definitions is shown to imply the axioms. The present results apply to the familiar von Neumann version of the ordinals, but the methods used are easily adapted to other versions. ? 1. Axioms for ordinal theory. According to the usual interpretation, each ordinal is the set of all lesser ordinals. Thus zero is the empty set A, a successor ordinal x' is x U {x} for some ordinal x, and a limit ordinal is a set w of ordinals that is equal to its own sum set Zw. In fact, the class On of all ordinals might be characterized informally as the smallest class Z such that (1) AEZ, (2) (y) (yEZ D y'EZ), and (3) (w) (1w = w c Z D wEZ). More simply, On can be characterized as the smallest Z such that
10510054
Words in the History of a Turing Machine with a Fixed Input
The usual decision (halting or printing) problems for Turing machines (T.M.) are relative to an infinite class of them or an infinite class of inputs or1 a single T.M. We consider now a new type of problem (the consequence problem) con-neeted with a single input on a single T.M. It is known that in a definite sense everything which an erasing T.3{. can do can also be done by a nonerasing T.M., e.g., the halting problem is undeeidable in both bases. It turns out that for the consequence problem erasing does make an essential difference: decidable without erasing, but undecidable Mth erasing. From the undeeidability of the consequence problem for a T.M., it also follows that the same problem is undeeidable for certain simple monogenie normal systems. Consider a fixed finite alphabet A including a symbol b for the blank squares on a tape. For definiteness, we may assume that A = {b, ,, a}. Assume some familiar formulation of Turing machines. Definition. The word at any moment t in the history of a T.M. is the string consisting of the contents of the squares in the minimum block on the tape at t that includes all the marked squares and the square scanned at the initial moment (the origin). THEOREM 1. For arty fixed (finite) word at the initial moment, we can find a T.M. such that the set of words in its subsequent history is not ~cursive. Ullian shows [1] that there exists some nonrecursive but recursively enumerable set S such that there is a general recursive function f such that S = {f(0), f(f(O)), f(f(f(O))),-.. }. We can now give a T.M. which first erases the finitely many marks on the initial tape and returns to the origin, then it puts down the representation of 0 on tape, calculates the value of f(0), erasing everything else except the representation of f(0), calculates f2(0). In particular, we can do it in such a way that the symbol a is used, after the initial tape content is erased, only to mark f(0), f2(0), etc; it is erased before we evaluate f(f (0)) from f(0). Moreover, the string of a followed by the representation of f(0) always begins at the origin. Clearly the set of words in the history of this T.M. cannot be recursive because otherwise the set S would be recursive, whether a 'gd(0) ' belongs to the …
1953614
Alternative Proof of a Theorem of Kleene
11313688
Existence of Classes and Value Specification of Variables
In mathematics, when we want to introduce classes which fulfill certain conditions, we usually prove beforehand that classes fulfilling such conditions do exist, and that such classes are uniquely determined by the conditions. The statements which state such unicity and existence of classes are in mathematical logic consequences of the principles of extensionality and class existence. In order to illustrate how these principles enable us to introduce classes into systems of mathematical logic, let us consider the manner in which Godel introduces classes in his book on set theory. For instance, before introducing the definition of the non-ordered pair of two classes Godel puts down as its justification the following two axioms: By A4, for every two classes y and z there exists at least one non-ordered pair w of them; and by A3, w is uniquely determined in A4.
15004157
A Semantic QoS-Aware Discovery Framework for Web Services
Augmenting web services with explicit semantics forms the foundation of Service Oriented Architectures (SOAs) automation. As more and more Semantic Web Services (SWSs) are deployed, similar SWSs could have quite different quality-of-service (QoS) levels. The QoS-aware discovery becomes an important challenge. While some efforts try to solve it via Constraint Programming (CP), they suffer from the purely syntactic matchmaking method. Furthermore, the construction of constraints and the selection of services are completely dependent on the literal translation from QoS descriptions, which increase obstacles to actually apply CP. In this paper, we propose a semantic QoS-aware framework for SWSs discovery by combining the semantic matchmaking and CP. Initially, a QoS ontology is presented to define QoS data into service descriptions. Then the ontology reasoning is adopted to change previous syntactic matchmaking into a semantic way. Through confirming the compatibility of concepts, complex QoS conditions are solved as constraints and a selection algorithm is proposed to obtain the optimal offer. Finally, the prototype implementation of our framework is discussed and a SWSs discovery case is used to illustrate the comprehensive discovery process.
24555513
Semantic Annotation for CSSCI Academic Resources Based on Ontology
The paper emphatically elaborates the process and method of realizing semantic annotation for domain data based on the CSSCI conceptualization which constructed before. The essence is that extracting the academic resource instances from CSSCI data under the guidance of the object-oriented knowledge organization model, setting values for properties of the instances by methods of directly extracting based on relational data model, directly statistic calculating assisted by TF-IDF algorithm, synthesis algorithm combined criterion weighting with formal concept analysis and so on, discovering and analyzing semantic association between instances, so as to integrate all CSSCI academic resources into a huge knowledge map based on ontology and to support the knowledge retrieval and citation analysis.
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Research on Positioning Technology of Xinjiang Mobile GIS
In this paper, the characteristics of Xinjiang mobile communication network are analyzed, the results show that for a reliable mobile communication network, BS (base station) and base station control BSC (base station controller) monitoring system, cable and fiber optic cable monitoring system and mobile communications troubleshooting and dispatching system should be built. Combined with GIS, GPS, GSM and geographic database technology, positioning technology for different targets and detection on fault point are studied, the results indicate that precise positioning of static target can accurately locate the failure telecom equipment in the electronic map and provide data source for the realization of real-time monitoring and management system for telecom equipment based on GIS & wireless sensor network technology. GSM-based wireless messaging communications and real-time fast positioning of dynamic target is the key to the emergency repair vehicles monitoring and dispatch system. The proposed fault determination and fault point location method can provide fast accurately fault devices location with detailed geography information and prompt the efficiency of fault management and improve the reliability of telecom network services. The above research results are important basis on the design and implementation of the system.
206827796
Application of Support Vector Machine to Mobile Communications in Telephone Traffic Load of Monthly Busy Hour Prediction
Telephone traffic of busy hour is one of indicators of load capacity of telecommunication network, which has a significant meaning to dilate and modify the network. A good performance of predicting the monthly busy hour traffic load is cared about by the mobile operators. As a promising learning theory,support vector machine (SVM) has been studied and applied in a wide area, such as financial markets and weather forecast. In this paper, we use SVM to forecast monthly busy hour traffic load of two regions in Xinjiang. A good result has been achieved via an improved grid search method for the search of hyper-parameter of SVM.
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Reflections on Kurt Gödel
Newton/Descartes. Einstein/Gdel. The seventeenth century had its scientific and philosophical geniuses. Why shouldn't ours have them as well? Kurt Gdel was indisputably one of the greatest thinkers of our time, and in this first extended treatment of his life and work, Hao Wang, who was in close contact with Gdel in his last years, brings out the full subtlety of Gdel's ideas and their connection with grand themes in the history of mathematics and philosophy.The subjects he covers include the completeness of elementary logic, the limits of formalization, the problem of evidence, the concept of set, the philosophy of mathematics, time, and relativity theory, metaphysics and religion, as well as general ideas on philosophy as a worldview. Wang, whose reflections on his colleague also serve to clarify his own philosophical thoughts, distinguishes his ideas from those of Gdel's and on points of agreement develops Gdel's views further.The book provides a generous array of information on and interpretation of the two main phases of Gdel's career - the years between 1924 and 1939 at the University of Vienna, which were marked by intense mathematical creativity, and the period from 1940 to his death in 1978, during which he was affiliated with the Institute for Advanced Studies in Princeton, a time in which Gdel's interests steadily shifted from questions of logic to metaphysics. And it also examines Gdel's relations with the Vienna Circle, his philosophical differences with Carnap and Wittgenstein, the intimate and mutually fruitful friendship with Einstein, and the periodic bouts of depression for which Gdel was hospitalized a number of times over the course of his life.Hao Wang is Professor of Logic at The Rockefeller University and author of scores of articles and several books on logic, computers, and philosophy, including From Mathematics to Philosophy (extensively discussed with Gdel and containing contributions by him) and Beyond Analytic Philosophy: Doing Justice to What We Know (MIT Press Bradford Books). He is currently preparing a companion volume, Conversations with Kurt Gdel which will concentrate on Gdel's unpublished ideas. A Bradford Book.