Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of (...) the Continuum Hypothesis, is a classic of modern mathematics. The continuum hypothesis, introduced by mathematician George Cantor in 1877, states that there is no set of numbers between the integers and real numbers. It was later included as the first of mathematician David Hilbert's twenty-three unsolved math problems, famously delivered as a manifesto to the field of mathematics at the International Congress of Mathematicians in Paris in 1900. In The Consistency of the Continuum Hypothesis Gödel set forth his proof for this problem. In 1999, Time magazine ranked him higher than fellow scientists Edwin Hubble, Enrico Fermi, John Maynard Keynes, James Watson, Francis Crick, and Jonas Salk. He is most renowned for his proof in 1931 of the 'incompleteness theorem,' in which he demonstrated that there are problems that cannot be solved by any set of rules or procedures. His proof wrought fruitful havoc in mathematics, logic, and beyond. (shrink)

A study of the contribution semantics makes to the syntactic patterns of English: an intepretive theory of grammar.

8.3 The consistency proof -- 8.4 Applications of the consistency proof -- 8.5 Second-order arithmetic -- Problems -- Chapter 9: Set Theory -- 9.1 Axioms for sets -- 9.2 Development of set theory -- 9.3 Ordinals -- 9.4 Cardinals -- 9.5 Interpretations of set theory -- 9.6 Constructible sets -- 9.7 The axiom of constructibility -- 9.8 Forcing -- 9.9 The independence proofs -- 9.10 Large cardinals -- Problems -- Appendix The Word Problem -- Index.

According to the alternative semantics for focus, the semantic reflec of intonational focus is a second semantic value, which in the case of a sentence is a set of propositions. We examine a range of semantic and pragmatic applications of the theory, and extract a unitary principle specifying how the focus semantic value interacts with semantic and pragmatic processes. A strong version of the theory has the effect of making lexical or construction-specific stipulation of a focus-related effect in association-with-focus constructions (...) impossible. Furthermore, while focus has a uniform import, the sources of meaning differences in association with focus are various. (shrink)

Investigates the application of logic to problem solving and computer programming. Requires no previous knowledge in this field, and therefore can be used as an introduction to logic, the theory of problem-solving and computer programming. Annotation copyrighted by Book News, Inc., Portland, OR.

This volume tackles the slippery subject of 'meaning'.

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...) pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

Investigating meta-programming within the logic programming paradigm, Meta-Logics and Logic Programming presents original research on an important extension of logic programming that makes it more amenable for knowledge representation and programming in general. The 12 contributions, many written especially for this book, explore the foundations, language design issues, and applications of meta-programming in logic programming. Meta-programming—the process of writing computer programs that can manipulate representations of other programs—has been key both in the foundations of computer science and in its practical (...) developments. Examples of meta-programs include compilers, interpreters, program analyzers, and partial evaluators. The choice of logic programming as a basis for meta-programming offers several practical and theoretical advantages: among them, the possibility of tackling critical foundational problems of meta-programming within a strong theoretical framework, and the surprising ease of programming. The usual framework of logic programming (and more generally first-order logic), however, has to be modified and extended to formally deal with meta-programs, extensions the editors call "meta-logics." Along with an exploration of meta-programming in logic programming, the definitions, formal properties, and use of these extensions constitute one of the book's main themes. The first part of the book, Foundations, focuses on the representation problem—how object programs are represented within meta-programs. The second part, Language Support for Meta-Logics, is concerned with language extensions that make meta-programming easier and more elegant. The third part, Meta-Logics for Knowledge Management, deals with the use of meta-logic for advanced knowledge representation purposes. Logic Programming series. (shrink)

Expressions in a language, whether words, phrases, or sentences, have meanings. So it seems reasonable to suppose that there are meanings that expressions have. Of course, it is fashionable in some philosophical circles to deny this.

Classical logic, beginning with the work of Aristotle, has developed into a powerful and rigorous mathematical theory with many applications in mathematics and ...

Previously published: Princeton University Press, 1940.