Book Description

Volume II, on formal (ZFC) set theory, incorporates a self-contained "chapter 0" on proof techniques so that it is based on formal logic, in the style of Bourbaki. The emphasis on basic techniques provides a solid foundation in set theory and a thorough context for the presentation of advanced topics (such as absoluteness, relative consistency results, two expositions of Godel's construstive universe, numerous ways of viewing recursion and Cohen forcing). ... Read more

Reviews (1)

Excellent book as an advanced introduction.
My opinion is that this is the most readable and user friendly advanced book on Set Theory today. Especially the treatment of (Cohen's) Forcing is valuable, since other expositions on this subject were not suitable for beginning graduate students (Kunnen - very good, but very sophisticated, Jech - also good, but presents Forcing via Boolean Algebra, which is less widely used today, and the book is too long as an advance introduction).
Also the style or writing is inviting, and not intimidating.
Very significant book, pedagogically. ... Read more

Book Description

This volume is a translation of the book Gödel, written in Japanese by Gaisi Takeuti, a distinguished proof theorist. The core of the book comprises a memoir of K Gödel, Takeuti's personal recollections, and his interpretation of Gödel's attitudes towards mathematical logic. It also contains Takeuti's recollection of association with some other famous logicians. Everything in the book is original, as the author adheres to his own experiences and interpretations. There is also an article on Hilbert's second problem as well as on the author's fundamental conjecture about second order logic. ... Read more

Book Description

This graduate-level text offers a theoretical treatment of the fundamental concepts and methods of automated deduction. By presenting an account which covers resolution theorem-proving in order-sorted first-order logic it provides a self-contained account suitable for students coming to the subject for the first time. Both Gentzen-style sequent calculi and the refutation method known as resolution are treated in detail. Various strategies for pruning resolution search spaces, such as linear, hyper- and ordered resolution are covered. Numerous examples are presented to illustrate the examples discussed. As a result students will find this a readily accessible introduction to this subject. ... Read more

Reviews (4)

How to reuse some older ideas
Acquired this book many years ago (early 199x). It was inspiring to read about efforts to produce dedicated hardware for cellular automata. Recently Stephen Wolfram's a new kind of science made me read this older book again. Then it was interesting to write a cellular automata program in C++ and afterwards in a mixed Python/C version. The rules to use were taken from this particular book. The general hardware nowadays has enough power to even program in a higher level language and still show adequate performance. In short: the book is inspiring and the presented rules can be used to investigate cellular automata.

Avoid
I came to this book after having seen it referenced in a number of more general books with chapters on cellular automata. The book describes numerous experiments in varying levels of detail that are suitable for running on cellular automata. The example experiments are both abstract ('Life' or 'Parity' rules) and concrete (simulation of physical systems). Little attention is paid to simulating either social or biological processes. There is a lot here but the level of detail is poor. Many explanations of the simulations assume considerable knowledge of the physical systems being simulated or the weaknesses of other approaches that may used. The greatest weakness of this book is that it relies on a specific hardware implementation of a CA machine. All examples require this machine (a PC card) to be installed in your PC in order to run. This card appears to no longer be available (no supplier details are given in the book). The use of Forth (a language using reverse polish notation) to describe rules and the reliance on specific hardware makes the rules discussed difficult to implement on a software CA. The rules presented in the second half of the book are so tied to the hardware implementation of the unavailable PC card as to be impossible to translate directly into English-like rules. This is a definite dud.

buy for the ideas, not the implementation
This is a terrific book that takes a step-by-step approach to cellular automata, especially for modelling. Within the first two chapters I had already found several interesting ideas for improving my own general-purpose automata program.

The part of the book that is most dated is the discussion of a specific hardware card and software designed for IBM PCs and ATs, and a specific dialect of Forth that can be used to program automata that will run on this card. Obviously this is no longer the mainstream approach to programming automata - even massively parallel systems programming has moved away from Forth. For me, I think of it as pseudo-code instead of a program example, and the book is still very very useful.

So on the whole, I would say this is a valuable addition to the bookshelf of any automata enthusiast.

Oh my god, what a book!
This book is absolutely stunning. It's a little dated, but the classes of problems that Toffoli and Margolus attack with dirt-simple machines is completely astonishing. I guarantee that you will not be able to put it down -- which is odd to say about a computer book. Read it! ... Read more

Book Description

This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of first-order logic formalization. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic, logic programming theory, category theory, modal logic, linear logic, first-order arithmetic and second-order logic. In each case the authors illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. For the new edition, they have rewritten many sections to improve clarity, added new sections on cut elimination, and included solutions to selected exercises. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. ... Read more

Reviews (1)

You Call Yourselves Analytic Philosophers?
That this book is not yet a foundational text in American graduate programs for analytic philosophy (or mathematics or computer science, for that matter) says something about an intellectual boundary those disciplines do not want to openly cross. Troelstra and Schwichtenberg, two hardened proof theorists with a mathematical conception of the issues involved in proof theory, cover nearly all of that subdiscipline's 70-year history in this affordable volume; and while it's not the clearest book ever written on the subject, it's a damn sight less obfuscatory than the Springer and North-Holland "monographs" (perhaps) available on the topic.

It's absolutely no fun to read at all, but it is resolutely Bourbakist in its avoidance of "proprioprecocity"; you will learn nothing that will impress others from reading this book, but you will also learn that intellectual techniques and ideas you thought were "cutting-edge" should be disquoted. And although the deflation of "minimalist" accounts of truth awaits an ill-tempered model theorist able to write competent English, this book contains contains a future of sorts for analytic thought; circling around the issues raised by computational complexity, as manifested in thought and action.

If this future be unappealing (as you might reasonably think), perhaps analysis as a mode of thought should be somewhat "deprivileged", such that our description of acceptably orderly cognition is brought more into line with the wanton reasoning practices of "natural consciousness". But at any rate: here is a bar, here raise. ... Read more

Book Description

A certain category of infinite strings of letters on a finite alphabet is presented here, chosen among the 'simplest' possible one may build, both because they are very deterministic and because they are built by simple rules (a letter is replaced by a word, a sequence is produced by iteration). These substitutive sequences have a surprisingly rich structure.The authors describe the concepts of quantity of natural interactions, with combinatorics on words, ergodic theory, linear algebra, spectral theory, geometry of tilings, theoretical computer science, diophantine approximation, trancendence, graph theory. This volume fulfils the need for a reference on the basic definitions and theorems, as well as for a state-of-the-art survey of the more difficult and unsolved problems. ... Read more

Book Description

Book Description

Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part II supplements the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Gödel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Gödel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index.
... Read more

Reviews (1)

Not for the autodidact
Ten years ago, I took an undergraduate course in symbolic logic. Wishing recently to refresh my (extremely rusty) memories of the propositional calculus and the first-order predicate calculus, I picked up this meaty text and was extremely dismayed to find myself soundly defeated within the first few pages. Kleene does not even make a pretense of holding the reader's hand: either you get it or you don't. There is nothing even remotely "user-friendly" about this book's presentation of its material.

If one were to read this book under the guidance of a teacher, I think it might be worthwhile. It may not be fair for me to blame the author for my inability to understand his writing. If you're smarter than I am, you might breeze right through it.

I cannot recommend this book, though, good though it may be, for anyone who wishes to teach him/herself logic, nor for anyone who wishes to brush up on the subject. There are exercises for the reader to test his/her understanding of the material, but no answer key is provided. This is heavy-duty stuff, and not well-suited to the self-teacher. ... Read more

Reviews (2)

Finally -- Straight Talk About Incompleteness!
Well. This is the book. Read this instead of, or before you read Goedel's paper. Within 20 pages you will know the 'trick' that Goedel used. It's a beauty, but it is far easier to see it under Smullyan's tutelage than by coming to the classic paper cold, since Goedel uses a more difficult scheme to achieve his ends. Much work has been done since 1931, and we get the benefit of the stripping-down to essentials that such as Tarski (and Smullyan himself) have contributed.

The book has much of interest to those who wish to pursue the subject of the incompleteness and/or consistency of mathematics, or to come at Goedel from a number of angles. For me, though, the first 3 chapters were enough. I just wanted to find out how K.G. did what he did. Now I know, and I know where to go if I need even more.

The exercises are helpful to keep you on track and test your understanding. They also contribute materially to the exposition. A stumbling-block for many readers will be the extremely abstract nature of the discussion, and the new notations and definitions that constantly come at one. Viewing numbers as strings and strings as numbers (and knowing when to switch from one view to another) will be confusing at first. This is the hard part: what Goedel did, in essence, is demonstrate that one can view proofs in two ways ' as numbers, and as strings of characters. As in viewing an optical illusion, it is sometimes tough to hold the proper picture in mind.

Smullyan's book 'First-Order Logic' is enough preparation for this work. One must here, even more than there, keep straight the difference between the 'proofs' that are part of the subject matter (and so are strings of characters), and the proofs we go through that verify facts about these strings. Before we started reading this book, of course, we had some informal sense that we were going to prove something about proofs. What we are REALLY doing, though, is proving something about 'proofs'. You get the picture. Goedel must have been a lot of fun at parties.

Mainline Incompleteness with this Book!
I highly recommend this title because it supplys all the necessary proofs for a nuts and bolts understanding of incompleteness, including incompleteness proofs for Peano arithmetic and the unprovability of consistency.

This title is a difficult read but the only prerequisite is a familiarity of first-order logic equivalent to a one semester college course.

A lot of the proofs are based on new material and are easier to understand than the original work by KG.

An added benefit is the exercises. They are not impossible and aid in one's understanding.

This book is well worth the work in demands. ... Read more

Book Description

A book which efficiently presents the basics of propositional and predicate logic, van Dalen's popular textbook contains a complete treatment classical logic on the basis of Gentzen's Natural Deduction and the traditional two-valued semantics, culminating in the completeness theorems. The first chapter, containing a leisured treatment of propostional logic, is followed by an equally elaborate chapter on predicate logic. On the basis of the material of the first of two chapters the completeness theorem is established and an excursion is made into model theory. The main facts of model theory, e.g. compactness, Skolem-Loewenheim, elementary equivalence, non-standard models, quantified elimination and Skolem functions are covered in chapter Three. The exposition of classical logic is rounded off with a concise exposition of second-order logic. In view of the growing recognition of constructive methods and principles, one chapter is devoted to intuitionistic logic. This chapter contains a completeness proof for Kripke's semantics and a number of specific constructive features have been incorporated, e.g. a study of equality and apartness the disjunction and existence property, the Goedel translation. A new chapter has been added at the end of this edition, with the basics of the proof theory of natural deduction; derivations are studued for their own sake and weak normalisation is proved. A choise of exercises is added ranging from simple applications of the definitions to more sophisticated problems. ... Read more

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Can I fond answer book for this book ?
I am searching for answer book to "Logic and Structure&quot ... Read more

Book Description

The automatic verification of large parts of mathematics has been an aim of many mathematicians from Leibniz to Hilbert. While Gödel's first incompleteness theorem showed that no computer program could automatically prove certain true theorems in mathematics, the advent of electronic computers and sophisticated software means in practice there are many quite effective systems for automated reasoning that can be used for checking mathematical proofs. This book describes the use of a computer program to check the proofs of several celebrated theorems in metamathematics including those of Gödel and Church-Rosser. The computer verification using the Boyer-Moore theorem prover yields precise and rigorous proofs of these difficult theorems.It also demonstrates the range and power of automated proof checking technology.The mechanization of metamathematics itself has important implications for automated reasoning, because metatheorems can be applied as labor-saving devices to simplify proof construction. ... Read more

Book Description

Hardbound. Process Algebra is a formal description technique for complex computer systems, especially those involving communicating, concurrently executing components. It is a subject that concurrently touches many topic areas of computer science and discrete math, including system design notations, logic, concurrency theory, specification and verification, operational semantics, algorithms, complexity theory, and, of course, algebra.
This Handbook documents the fate of process algebra since its inception in the late 1970's to the present. It is intended to serve as a reference source for researchers, students, and system designers and engineers interested in either the theory of process algebra or in learning what process algebra brings to the table as a formal system description and verification technique. The Handbook is divided into six parts spanning a total of 19 self-contained Chapters.
The organization is as follows. Part 1, consisting of ... Read more

Book Description

"Experts in mathematical logic will find this book of engrossing interest. For mere philosphers it will have a different fascination: in seeing how the achievements of a genius can seem to him to provide a firm foundation for a species of Platonism and the conviction of the superiority of minds over computers, and at the same time can encourage him to favour a quasi-Leibnizian speculative metaphysics and theology. Hao Wang records and assesses the whole with an expert and balanced reasonableness." -- Sir Peter F. Strawson, Magdalen College, Oxford

Hao Wang (1921-1995) was one of the few confidants of the great mathematician and logician Kurt Gödel. A Logical Journey is a continuation of Wang's Reflections on Gödel and also elaborates on discussions contained in From Mathematics to Philosophy. A decade in preparation, it contains important and unfamiliar insights into Gödel's views on a wide range of issues, from Platonism and the nature of logic, to minds and machines, the existence of God, and positivism and phenomenology. The impact of Gödel's theorem on twentieth-century thought is on par with that of Einstein's theory of relativity, Heisenberg's uncertainty principle, or Keynesian economics. These previously unpublished intimate and informal conversations, however, bring to light and amplify Gödel's other major contributions to logic and philosophy. They reveal that there is much more in Gödel's philosophy of mathematics than is commonly believed, and more in his philosophy than his philosophy of mathematics. Wang writes that "it is even possible that his quite informal and loosely structured conversations with me, which I am freely using in this book, will turn out to be the fullest existing expression of the diverse components of his inadequately articulated general philosophy." The first two chapters are devoted to Gödel's life and mental development. In the chapters that follow, Wang illustrates the quest for overarching solutions and grand unifications of knowledge and action in Gödel's written speculations on God and an afterlife. He gives the background and a chronological summary of the conversations, considers Gödel's comments on philosophies and philosophers (his support of Husserl's phenomenology and his digressions on Kant and Wittgenstein), and his attempt to demonstrate the superiority of the mind's power over brains and machines. Three chapters are tied together by what Wang perceives to be Gödel's governing ideal of philosophy: an exact theory in which mathematics and Newtonian physics serve as a model for philosophy or metaphysics. Finally, in an epilog Wang sketches his own approach to philosophy in contrast to his interpretation of Gödel's outlook. ... Read more

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Meet Gödel the philosopher
Many mathematicians know about Gödel's famous theorem. But very few know about Gödel the man. Through this book, we come to know the man, especially Gödel the philosopher.

Through this book we find out that although Gödel and Einstein were close friends, Gödel, unlike Einstein, shunned public debate. He held philosophical views which he knew would be very controversial if he were to publicize them, and he greatly disliked publshing anything he could not prove rigorously. Accoringly, he instructed his biographer to publish these viewpoints only after his death.

This book contains hundreds of quotations from Gödel's conversations with the author. Fortunately, the author left in quotations that he he said he did not understand, trusting that others might.

Here are a few quotes:

"Consciousness is connected with one unity. A machine is composed of parts."

"The brain is a computing machine connected with a spirit."

"Materialism is false."

"Our total reality and total existence are beautiful and meaningful . . . . We should judge reality by the little which we truly know of it. Since that part which conceptually we know fully turns out to be so beautiful, the real world of which we know so little should also be beautiful. Life may be miserable for seventy years and happy for a million years: the short period of misery may even be necessary for the whole."

If you find Gödel's theorem interesting, I hope you will read this book and found out more about the man behind the theorem. ... Read more

Book Description

This is the answer key that supports the Intermediate Logic (Student). ... Read more

Book Description

Rotman, Joseph, Journey into Mathematics: The Worldof Proof Prompting readers to do mathematics, not merely readabout it, this proactive book has users reading and writing proofs at theoutset. Complete proofs are given from the start, and coverage begins withelementary mathematics to allow focus on the writing and reading of proofswithout the distraction of absorbing new ideas simultaneously.KEYTOPICS:Contains material that is familiar to calculus readers (i.e.,induction, binomial theorem, polygonal, using the Diophantine parametrizationof the circle by rational functions to solve trigonometric identities, etc.),and engages readers throughout with interesting expositions using history,etymology, and humorous asides. Material is presented as a 'single story' toaid in the subjects progressive growth and development.Formathematicians. ... Read more

Reviews (1)

A very entertaining read
The book has a very interesting approach to a wide range of topics in maths. Uses lots of anecdotes and historical facts to motivate the reader. I haven't really sat down to the exercises, but I love the stories and pointers.. sure shall get at the sources mentioned, sometime. ... Read more