Mathematical Logic. FOURTl-1 EI)ITJ()N. Elliott Mendelson. Queens College of the City University of New York. CHAPMAN & HALL. London· Weinheim · New. PDF | (New edition of the book - Edition added May 24, ) Hyper- textbook for students in mathematical logic. Part 1. Mathematical Logic. Helmut Schwichtenberg. Mathematisches Institut der Universitфt M№nchen. Wintersemester /
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We start with a brief overview of mathematical logic as covered in this course. course we develop mathematical logic using elementary set theory as given. Propositional logic is a mathematical system for reasoning about propositions and how they relate to one another. ○ Propositional logic enables us to. ○. In this text we study mathematical logic as the language and deductive system Mathematical logic introduces the notion of a formal proof: a finite sequence.
Leary, Lars Kristiansen, , pp, 1. The Game of Logic by Lewis Carroll, Hack, Hack, Who's There? Introduction to Logic by Michal Walickil, , pages, 4. Introduction to Mathematical Logic: A problem solving course by Arnold W. Nipkow, L. Paulson, M. Wenzel, , pages, 1. Lectures on Linear Logic by A. Logic for Computer Science by Jean H. Gallier, , pages, 1.
Logic for Computer Science Wikibooks, , online html. Logic For Everyone by Robert A. Logics of Time and Computation by Robert Goldblatt, , pages, 6. Mathematical Logic by Stephen G.
The general character of the book is clear from the foreword written by Barwise. This book has been written by mathematicians for mathematicians. The editor describes the Handbook as 'an attempt to share with the entire mathematical community some modern developments in logic' vii; my italics.
Given this outlook, it is no sur- prise that other people with a legitimate interest in logic, such as philosophers, linguists and computer scientists, will not find the book tailored to their needs. The style throughout nearly all the book is the technically elegant, clean but formalistic style of modern mathematics.
The philosopher or for that matter, computer scientist with modest logical background will undoubtedly find the book hard to read. Although the book is a 'Handbook' a word which conjures a picture of the handy compendia to be found in engineers' offices , virtually all the 'applications' of logic considered are to other branches of pure mathematics.
Applications of logic to foundational issues and to the practice of computation are mentioned only tangentially. The book is divided into four parts, corresponding to the now stan- dard division of logic into model theory, set theory, recursion theory and proof theory constructive mathematics is lumped with proof theory. The list of topics is notable for the omission of non-classical logics with the exception of intuitionistic logic.
This omission reflects the current lack of interest in this topic among North American logicians. I may add that in revenge for this omission the non-classical logicians under the leadership of Gabbay and Guenthner are preparing an even more This content downloaded by the authorized user from The section on model theory, edited with the cooperation of H. Keisler, is of a consistently high standard.
After a breezy run-through of first-order logic written by the editor, Keisler provides a useful survey of standard model theory; this chapter is in effect a scaled down version of the author's well known text co-authored with Chang. There follow two chapters by Eklof and Macintyre on ultraproducts and model com- pleteness, notable for their clear, unhurried exposition and useful motivating remarks. A superb expository article by Stroyan on Robinson's infinitesimal analysis is next.
The first part of this chapter should be accessible to the philosopher with average logical background; the more difficult second half shows how Gauss's famous memoir on curved surfaces can be given a very natural setting in the modernised calculus of infinitesimals. The preceding chapters in the model theory section are all bread-and- butter model theory, aimed at the algebraist or classical analyst of slight logical background.
The remaining chapters are more speculative. The first is a fine exposition by Makkai of the recent theory of admissible sets. In this theory, an abstract theory of definability is used to give a common framework for infinitary logic and generalized recursion theory. Following a basic insight of Kreisel, the theory generalizes not only the concept of 'recursive set' but also that of 'finite set' to an abstract setting. This permits a generalization of the compactness theorem to languages with expressions of infinite length.
So far the mathematical ap- plications of this theory have not been striking, but the theory should be of interest to philosophers as a case study of the fruitfulness of finding the right kind of generalization in mathematics.
The last chapter, by Joyal and Reyes, sketches the category-theoretical approach to model theory. As usual in category theory the disappearance of 'objects' allows one a clearer view of the underlying patterns in many model-theoretic constructions shorn of the nitty-gritty details of explicit languages and set-based structures. Model theorists have been slow to adopt category- theoretic methods, but as Macintyre's article shows, there are con- siderable gains in clarity and insight to be had from learning the language.
After two easy-going, chatty articles by Shoenfield and Jech on the axioms of set theory and the axiom of choice, there follow chapters by Kunen, Burgess, Devlin, Mary Ellen Rudin and Juhasz on in- finitary combinatorics, forcing, constructibility, Martin's axiom and set- theoretical topology. All of these articles are packed densely with results, mostly accompanied by complete proofs.
The article by Burgess, for ex- ample, is a meaty compendium of forcing techniques written in a com- pact and lucid style. Following a modern trend, the proof that forcing works is omitted so that varied applications of the method can be presented. Technically, the section on set theory is superb. However, it will not satisfy a philosopher interested in learning about the foundational aspects of the area.
Of course, this reflects the current situation in the subject. From a purely formal point of view, all the big classical pro- blems have been solved, mainly by the combination of the flexible machinery of constructibility and forcing. However, it can hardly be said that we have made great progress towards settling such basic questions as the continuum hypothesis.
Only one axiom is known which has some plausibility and settles the question, Godel's axiom of constructibility, which says that the universe of sets can be obtained by repeatedly form- ing the family of first-order definable subsets of the collection of sets so far generated.
This axiom is enormously powerful, and allows the definitive solution of most classical problems, including a positive solu- tion to the Generalized Continuum Hypothesis. Godel himself originally recommended adoption of the axiom, as providing a sort of completion' to the axioms of set theory; however, he later changed his mind.
Most set theorists seem to follow him in this, though for reasons that are obscure to me. This leaves the foundations of the subject in an unsatisfactory state.
Set theorists in their daily work adopt an agnostic attitude, so that they can tell their topologist colleagues for example , Well, if you adopt Martin's axiom and not CH, then However, there seems to be a general tenden- cy towards a belief in the existence of enormously large sets. The reasons for this are again unclear to me, but the general principle seems to be that the universe of sets should be as large and 'hairy' as possible. Uncoun- table measurable cardinals are especial favourites.
The existence of these cardinal numbers can not it seems be justified by the idea that the set- theoretical universe has no conceivable end - the usual justification for large cardinal axioms. Rather, the justification seems to be the vague This content downloaded by the authorized user from However that may be, these cardinals are certainly 'hairy' enough.
On the whole it seems a pity that Godel did not stick to his original opinion. We say that Zermelo proved the well ordering theorem in - but this only means that he boldly postulated the axiom of choice and stuck to it in the face of determined opposition. If Godel had done the same thing for the axiom of constructibility, we would very likely say that he proved the Generalized Continuum Hypothesis in The ax- iom of constructibility has a great deal to recommend it; if it were adopted, all we would miss would be the monstrous cardinal numbers mentioned above.
Properties of first-order theories. Additional metatheorems and derived rules. Rule C.
Completeness theorems. First-order theories with equality. Definitions of new function letters and individual constants. Prenex normal forms. Isomorphism of interpretations. Categoricity of theories. Generalized first-order theories. Completeness and decidability. Elementary equivalence. Elementary extensions. Nonstandard analysis. Semantic trees. Chapter 3: Formul Number Theory. Axiom system.
Number-theoretic functions and relations. Primitive recursive and recursive functions. Godel numbers. The fixed point theorem. Recursive undecidability.
Chapter 4: Axiomatic set theory. An axiom system. Ordinal numbers. Finite and denumerable sets.
Initial ordinals. Ordinal arithmetic. The axiom of choice.