21405683
Set-Theoretical Basis for Real Numbers
twenty axioms used by Tarski which are sufficient for the arithmetic of real numbers and are to the effect that real numbers form a complete ordered field.2 Indeed, we cannot prove in L all Tarski's twenty axioms in their full generality. One of them, stating in effect that every bounded class of real numbers possesses a least upper bound, can only be proved as a metatheorem which states that every bounded nameable class of real numbers possesses a least upper bound. However, all the other nineteen axioms can be proved in L without any modification. This result may be of some interest because the axioms of L are considerably weaker than those commonly employed for the same purpose. In L variables need to take as values only classes each of whose members has no more than two members. In other words, only classes each with no more than two members are to be elements. On the other hand, it is usual to assume for the purpose of natural arithmetic that all finite classes are elements, and, for the purpose of real arithmetic, that all enumerable classes are elements. 2. The system L. In developing L, elementary logic (viz., the theory of truth functions and quantifiers) is assumed from the beginning. The single additional primitive is the two place predicate e with which we can generate formulas of L from variables and logical operators in the usual manner. The usual definitions (e.g., D9-D35 of [21) of the theory of classes and relations are assumed. In particular, let tx denote the unit class of x, x u y denote the sum class of x and y, and V denote the class of all elements. The axioms of L are given by the following three statements: Mi. tx U by f V. M2. x = y D . x e z D y e z. M3. If q is any formula of L in which y is not free, then . (3y)(x)(x e yW
20547783
The Formalization of Mathematics
Zest for both system and objectivity is the formal logician's original sin. He pays for it by constant frustrations and by living ofttimes the life of an intellectual outcaste. The task of squeezing a large body of stubborn facts into a more or less rigid system can be a painful one, especially since the facts of mathematics are among the most stubborn of all facts. Moreover, the more general and abstract we get, the farther removed we are from the raw mathematical experience. As intuition ceases to operate effectively, we fall into many unexpected traps. The formal logician gets little sympathy for his frustrations. He is regarded as too rigid by his philosophical colleagues and too speculative by his mathematical friends. The life of an intellectual outcaste may be a result partly of temperament and partly of the youthfulness of the logic profession. The unfortunate lack of wide appeal of logic may, however, be prolonged partly on account of the fact that very little of the well-established techniques of mathematics seems applicable to the treatment of serious problems of logic. The axiomatic method is well suited to provide results which are both exact and systematic. How attractive would it be if we could get an axiom system in which all the axioms and deductions were intuitively clear and all theorems of mathematics were provable? Such a system would undoubtedly satisfy Descartes who admits solely intuition and deduction, which are, for him, the only “mental operations by which we are able, wholly without fear of illusion, to arrive at the knowledge of things.” Indeed, according to Descartes, intuition and deduction “are the most certain routes to knowledge, and the mind should admit no others. All the rest should be rejected as suspect of error and dangerous.”
26896458
The Axiomatization of Arithmetic
9428173
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
33404030
Some Facts About Kurt Godel
14517479
The Logic of Automata—Part I
We are concerned in this paper with the use of logical systems and techniques in the analysis of the structure and behavior of automata. In Section 2 we discuss automata in general. A new kind of automaton is introduced, the growing automaton, of which Turing machines and self-duplicating automata are special cases. Thereafter we limit the discussion to fixed, deterministic automata and define their basic features. We give methods of analyzing these automata in terms of their states. Four kinds of state tables--complete tables, admissibility trees, characterizing tables, and output tables--are used for this purpose. These methods provide a decision procedure for determining whether or not two automaton junctions behave the same. Finally, a class of well-formed automaton nets is defined, and it is shown how to pass from nets to state tables and vice versa. A coded normal form for nets is given. In Section 3* we show how the information contained in the state tables can be expressed in matrix form. The (i, j) element of a transition matrix gives those inputs which cause state S~ to produce state Sj. Various theorems are proved about these matrices and a corresponding normal form (the decoded normal form or matrix form) for nets is introduced. In Section 4* we first show how to decompose a net into one or more subnets which contain cycles but which are not themselves interconnected cyclically. We then discuss the relation of cycles in nets to the use of truth functions and quantifiers for describing nets. We conclude by relating nerve nets to other automaton nets.
5067889
Facial Expression Synthesis and Recognition with Intensity Alignment
A method of cooling section steel, particularly sectional steel girders, from rolling heat, wherein, prior to a final air cooling, a water cooling process is carried out in such a way that section steel portions with material concentrations are cooled at outwardly projecting portions of the steel sections over a width of application and a duration of application which are variable and subject to a predetermined cooling strategy supported by a process computer to a value which is at least still slightly above the transformation temperature Ar1.
6971305
Video streaming over multi-hop wireless networks
In this paper, we considered the problem of transporting layered video over erroneous multi-hop wireless networks and proposed a distributed system. Our proposed distributed scheme is comprised of distributed control (DC), distributed buffer (DB) and distributed error control schemes. The DC scheme improves the efficiency of the network bandwidth usage and reduces the end-to-end delay of the streaming application. End-to-end delay jitter can be reduced by proper use of the DB nodes' buffer. Replacing the traditional EEC and ARQ with our distributed FEC and ARQ scheme reduces the error-protection overhead and ARQ delay and improves the wireless channel throughput. A layered video module is developed in GlomoSim 2.03 for video streaming application. We simulated our distributed scheme employing the layered video module in GlomoSim 2.03. Our simulation results confirm that QoS of streaming video over erroneous multi-hop wireless network can be improved using our proposed distributed scheme.
8395605
FLTL-MC: Online High Level Program Analysis for Web Services
Although Java- or .NET- centric technologies are the most commonly used in Web services, they are by no means the only ones in practice. This paper proposes an online finite model checking tool FLTL-MC to check the high level safety and liveness properties in complex distributed web service systems, which can offer both a richer and more natural way to search errors. Liveness properties can specify desirable system behaviors which must be satisfied eventually, but are not always satisfied. Existing software model checkers cannot verify liveness in real code because doing so requires finding an infinite execution that does not satisfy aliveness property. In our proposed model, we adopt the finite linear temporal logic to specify the semantics of the online model checking, use binary instrumentation to obtain the distribute states and apply the FLTL-MC engine to dynamically verify the finite linear temporal logic properties in Web service systems. At last, we investigate the well-known distributed protocol WS-Reliable Messaging to demonstrate its applications and detect some hidden bugs with our prototype system.
14513982
An Online Model Checking Tool for Safety and Liveness Bugs
Modern software model checkers are usually used to find safety violations. However, checking liveness properties can offer a more natural and effective way to detect errors, particularly in complex concurrent and distributed e-business systems. Specifying global liveness properties which should always eventually be true proves to be more desirable, but it is hard for existing software model checkers to verify liveness in real codes because doing so requires finding an infinite execution. For solving such a challenge, this paper proposes an online checking tool to verify the safety and liveness properties of complex systems. We adopt the linear temporal logic to describe the semantics of the finite model checking, use binary instrumentation to obtain the distribute states and apply a checking engine to dynamically verify the finite trace linear temporal logic properties. At last, we demonstrate the method in a distributed system using distributed protocol Paxos and achieve good results by experiments.