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Mathematical Logic (Oxford Texts in Logic)
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If something is to be a reason to detract, say, one star tops, then let it be the price. Indeed, for the sheer material aspect of the book (number of pages, paper quality, etc) it seems there is no good reason for it to cost $62.00 bucks, on the other hand the actual logical material presented in the book is so didactic, nicely presented along snippets about key concepts and logicians, that it worths the price paid twice! No other book can compare to this one in terms of efficiency, being the sole complain that it is too short.
Maybe we can hope for another enlarged edition in the future to cover second order logic.
This is the ideal book for those who struggled with a Van Dalen or the like.
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.Read more ›