Mathematical Logic - Computability, Set Theory, Model Theory, etc

The skeleton of this narrative nonfiction book is a science fiction story without the usual mangling of physics. The flesh is composed of self-contained non-technical mainstream explanations and examples of the field of mathematics which deals with meaning, called Model Theory. Technical terms are avoided, except to occasionally serve as keywords.

This book provides an introduction to Model Theory which can be used as a text for a reading course or a summer project at the senior undergraduate or graduate level. It is also a primer which will give someone a self contained overview of the subject, before diving into one of the more encyclopedic standard graduate texts.

Model theory investigates the relationships between mathematical structures ('models') on the one hand and formal languages (in which statements about these structures can be formulated) on the other. Example structures are: the natural numbers with the usual arithmetical operations, the structures familiar from algebra, ordered sets, etc. The emphasis is on first-order languages, the model theory of which is best known.

This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions. It introduces the classic results, as well as more recent developments in this vibrant area of mathematical logic. Concrete mathematical examples are included throughout to make the concepts easier to follow.

This is an up-to-date textbook of model theory taking the reader from first definitions to Morley's theorem and the elementary parts of stability theory.

This book provides a very clear and well-developed approach. It covers axioms and set theory for the mathematics student on the upper undergraduate or graduate level. Throughout, emphasis is on axioms and theorems; proofs are informal.

This book is a brief introduction to the philosophy of set theory. It is written for students with a little background in logic, and some high school mathematics, aims to scratch the tip of the surface of the philosophy of set theory. By the end of this book, students reading it might have a sense of:

This book brings together several directions of work in model theory between the late 1950s and early 1980s. It contains expository papers by pre-eminent researchers. Part I provides an introduction to the subject as a whole, as well as to the basic theory and examples. The rest of the book addresses finitary languages with additional quantifiers, infinitary languages, second-order logic, logics of topology and analysis, and advanced topics in abstract model theory. Many chapters can be read independently.

Kurt Gödel’s famous First Incompleteness Theorem shows that, for any sufficiently rich theory that contains enough arithmetic, there are some arithmetical truths the theory can express. but cannot prove. How is this remarkable result established? This short book explains. It also discusses Gödel’s Second Incompleteness Theorem.

The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book -- extensively rewritten for its second edition -- will be accessible to philosophy students with a limited formal background. It will be of equal interest to mathematics students taking a first course in mathematical logic.

This book is an introduction to metamathematics and Gödel's Theorems. It covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related.

This book gives a thorough development of the foundations of computability, from the definition of Turing machines up to finite injury priority arguments. Key topics include relative computability, and computably enumerable sets, those which can be effectively listed but not necessarily effectively decided, such as the theorems of Peano arithmetic.

Computability Theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications.

A textbook on the semantics, proof theory, and metatheory of first-order logic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. It is based on the Open Logic project, and available for free download at slc.openlogicproject.org.