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What is Mathematical Logic?

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Although this book - What is Mathematical Logic? - is written in an informal and entertaining style, it is unlikely to appeal to a reader not familiar with predicate calculus, recursive functions, and set theory. Despite its innocuous title, this little book is surprisingly rigorous.

The six chapters are derived from a series of lectures given by the five authors - J. N. Crossley, C. J. Ash, C. J. Brickhill, J. C. Stillwell, and N. H. Williams - at Monash University and University of Melbourne in 1971. The lectures were substantially revised for publication.

Only the first chapter, a detailed historical survey of mathematical logic, can be readily appreciated by the non-mathematician. The remaining five chapters examine advanced topics in mathematical logic including the Godel-Henkin Completeness Theorem, Model Theory, Turing machines and recursive functions, Godel's Incompleteness Theorem, and advanced set theory.

Chapter 2 introduces the Godel-Henkin Completeness Theorem, a proof that predicate calculus is complete. Chapter 2 is not easy, but it is essential to acquire a reasonable familiarity with predicate calculus before moving forward.

Chapter 3 offers a detailed look at model theory, the study of relations between formal languages and the interpretation of formal languages. Topics include Predicate Calculus with Identity, the Compactness Theorem, and the Lowenheim-Skolem Theorems. I had substantial difficulty with the details, but I did gain a general understanding and appreciation for model theory.

Chapter 4 addressed in considerable detail a more familiar topic, Turing machines and recursive functions. The discussion concludes with a key proof: there is no algorithm which will enable us to decide, given any particular formula of predicate calculus, whether or not this particular formula is deducible from the axioms of predicate calculus.

Chapter 5 was a detailed examination of Godel's Incompleteness Theorem for formal systems that include arithmetic of the natural numbers. I had less difficulty with this topic as I had previously read Godel's Proof by E. Nagel and J. R. Newman. This chapter would very likely be tough going for a reader entirely new to Godel's exceeding complex and abstruse proof.

Chapter 6, titled Set Theory, might be better named Advanced Set Theory. I was entirely new to the Axiom of Choice and the Generalized Continuum Hypothesis.

I highly recommend this intriguing and lively look at mathematical logic to readers with some familiarity with this rather formidable subject. For readers new to mathematical logic, I suggest that the following books might be better starting points.

Foundations and Fundamental Concepts of Mathematics by Howard Eves is outstanding. The chapter titled Logic and Philosophy is an excellent introduction to mathematical logic.

The Advent of the Algorithm by David Berlinski is an eclectic, rather bizarre introduction to a complex mathematical topic. Although many reader reviewers aggressively criticize this book, I enjoyed puzzling my way through Berlinski's discursive discussions.

Godel's Proof by Ernest Nagel and James R. Newman offers a fascinating look at a mind boggling, incredibly complex, inventive mathematical proof.

After a 10-page historical survey of logic from the 1850s through the 1960s, similarly brief chapters on Completeness, Model Theory, Recursion Theory, the Incompleteness Theorems, and Set Theory give an idea of what might be covered in an undergraduate course and the first several graduate courses in mathematical logic. (The last 5 pages of the book are an introduction to forcing arguments and a fairly detailed sketch of the consistency of not-GCH.)

Results are clearly and carefully stated; and while sketches of proofs have a hard time staying nontechnical and still meaningful, most such attempts are admirable.

A marvel of brevity while not watering anything down.

This is an introduction to the main ideas and results of mathematical logic. It is primarily a text for non-logicians but it is still very serious. Practically everything is proved, and the proofs are carefully crafted and not too technical. For a reader with a bit of mathematical background this is far more valuable than the more typical logic-for-casual-readers books, such as for instance "Gödel's Proof" by Nagel & Newman, which are too chatty and trivial and don't really prove anything. By contrast, a high point of this book is a very accessible treatment of the proof of Gödel's incompleteness theorem in a matter of a few pages. On the other hand, this book is perhaps not chatty enough: the clear proofs and discussions of the main results are nicely done, but the discussions of historical background, motivation and context are very sketchy.

Despite its title, this book does not answer the question in the title as a novice would ask it. Readers who know only a little about logic would not have the concerns about logic that are addressed in this book. They would want to know what mathematical logic is, what it looks like in use, and whether they'd like to learn it in detail by reading a longer book after this one. They're as likely to be discouraged as inspired by what the authors tell them here.

The title should be something like: Sketches of Some Important Theorems in Mathematical Logic and Set Theory. As the Preface states: "Our aim was to introduce the very important ideas in modern mathematical logic without the detailed mathematical work which is required of those with a professional interest in logic." The book is concerned about the completeness theorem of predicate logic, cardinality of models and non-standard models in predicate logic with identity, Turing machines and unsolvability in computation and in predicate logic, Gödel's incompleteness theorems, and set theory, including a discussion of axioms, ordinals, cardinal numbers, constructible sets and the independence of the axiom of choice and the continuum hypothesis.

The problem is, if you don't already know much about logic, reading these chapters isn't going to make you informed in any practical, helpful way; and if you have studied logic enough to understand the formal context and thus the intellectual value of the theorems discussed, then you may have already encountered the theorems in some form elsewhere; and if you have encountered them before and you understand their importance, then you're probably ready for, or already had, a more detailed account than you'll get from this book.

For an accessible, non-technical look at the historical, mathematical concerns that are in the background to the development of logic and set theory, see Morris Kline's (1980) Mathematics: The Loss of Certainty. On Gödel, a classic, popular mathematics account of his incompleteness theorem is in Ernest Nagel and James R. Newman's (1958) Gödel's Proof, revised in 2001 by Douglas Hofstadter.