Mathematical Logic (Oxford Texts in Logic)

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Assuming no previous study in logic, this informal yet rigorous text covers the material of a standard undergraduate first course in mathematical logic, using natural deduction and leading up to the completeness theorem for first-order logic. At each stage of the text, the reader is given an intuition based on standard mathematical practice, which is subsequently developed with clean formal mathematics. Alongside the practical examples, readers learn what can and can't be calculated; for example the correctness of a derivation proving a given sequent can be tested mechanically, but there is no general mechanical test for the existence of a derivation proving the given sequent. The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich's theorem characterising the computably enumerable relations. Rigorous proofs of the adequacy and completeness proofs of the relevant logics are provided, with careful attention to the languages involved. Optinal sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch. Throughout the book there are notes on historical aspects of the material, and connections with linguistics and computer science, and the discussion of syntax and semantics is influenced by modern linguistic approaches. Two basic themes in recent cognitive science studies of actual human reasoning are also introduced. Including extensive exercises and selected solutions, this text is ideal for students in logic, mathematics, philosophy, and computer science.

Mathematical Logic (Oxford Texts in Logic) Review

Such an excellent text. I congratulate logician Wilfred Hodges whose works I have had the honour of studying. This is an excellent text in following sense.

A deep complex work in mathematical logic would be at least 600+ pages of pure mathematical reasoning. I did a Masters course with the brilliant Moshe Machover (A course in mathematical logic, which he wrote with John Bell).

But a rigorous account convering Quantificational Logic, Model Theory, Recursive Functions and large proofs (such as that of Putnam, Davis, Matyasevich's theorem), Formal Set Theory, Non-standard Analysis, would at lease be as long as Machover's book -- in fact, a long-hand explanation of what he did in the book, solutions of exercises would make the book at least 800 pages.

So then, if you don't have a career in logic and logic-related sciences in mind, you can get a great introduction here to logic by one of recent logic's icons. His Model Theory book is something of a bible.

Once you appetite is whetted for Logic by this book, you might next go to Category Theory/Topos approach to logic in the Lawvere's expose: conceptual mathematics (2nd Edition!).

From then on sky is the limit: You can tend to that limit via the work of Saunders Maclane in SHEAVES in Geometry and Logic. But for the non-mathematician, this book is ENTIRELY inaccessible.

A WORD or two of HOPE ! Mathematics is like a large tapestry and one mustn't be too fussy about the EXACT coverage of "everything" you want in a book. The point is that as you read an good book - -like Hodge's -- you will have learnt a portion of that tapestry. The next book will conver more of it, you can skip the parts you have read from Hodges or Lawvere's or find them more easily workable.

The most difficult part of doing logic/mathematics is ballancing the pace of working and the mind-set for studying it. I read a partial account of how in musical training in 18th century getting the mind to be in the right attitude was the central focus - which was part of the environment of creativity.

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