Introduction to Mathematical Logic By Elliott Mendelson

Summary Introduction to Mathematical Logic

Very interesting. Good, clear and intuitive explanations, without lapsing to lack of rigour. English A good reference English For twenty years we must stop this brain from working.

Or, Getting Over Mendelson

A course taught out of Elliott Mendelson's Introduction to Mathematical Logic was my undoing as a university student. I had taken a few logic courses in the philosophy department and done very well, but I wanted to 'get real' with a course for math majors; the instructor was kind as I went on about Tarskian satisfaction, but the crucial midterm was a sentence of death of sorts (at a later point he offered the consoling words too bad). I didn't stop with logic, though; much as becoming a writer in the Soviet Union famously involved pronouncing yourself one, logic is a topic that even the ill-calibrated are not generally begrudged an interest in. Though the verdict may be out on whether I actually ever learned to reason classically, I do not regret the time spent with it.

I have had a copy of the Fourth Edition of this book close to hand during this extended sojourn, and although it is one of the major mathematical logic texts I cannot say I have come to love it. The usual problem with famous works like Alonzo Church's Introduction to Mathematical Logic is that they leave out a great deal (Church did not see fit to include his own proof of the undecidability of the first-order predicate calculus in the Vol. 1 that saw print); Mendelson cannot be convicted on this charge, as all the major topics of 'intermediate logic' are covered in his textbook. In fact, as you work through the byways of the book you are surreptitiously introduced to a great deal of material (the P and NP problem is only one example) that officially falls outside the main line of the book.

Mendelson's 'pocket' introductions to things like model theory proper and recursion theory are good places to start with advanced logic, after you grasp the famous major metalogical results. And here, you see, is the problem; one begins to feel a great deal like Achilles facing down the tortoise as one tries to grasp completeness, incompleteness, and computability using Mendelson's explanantions. In fact, it must unfortunately be said that this book may be in a class with Walter Rudin's famous Principles of Mathematical Analysis as an introductory text you only grasp when you have learned the material out of other books. If logic deserves a wider audience, somebody should go about getting it one; this is the form it takes as an unimportant subdivision of mathematics accessible to very few.

Perhaps if you could only buy one logic book in your life this should be it; you can, however, read many others as well. This is fundamentally a repository of information, rather than a resource. The inclusion of a version of Gentzen's complicated consistency proof for arithmetic in this edition is a welcome addition to Mendelson's standard-setting appendix on second-order logic, but the added sections on modal logic and nonstandard models of arithmetic are of no great value (this is not a judgment on the importance of those topics, but the exposition). English There are more rigorous, professional introductions to the subject—but they are designed for people who already know a good deal about the subject, or are exceptional mathematicians. This is a fast-paced and thorough introduction, not for the faint of heart or casual student of mathematical logic. The text is clear and demanding, and provides all of the insight that one could reasonably hope for in the subject, as we understand it to date. It also leaves much of the work to the reader in the form of exercises at the end of each section and chapter.

You cannot read this if you aren't already proficient in both mathematical proofs and symbolic logic. English

This established standard covers the basic topics for a first course in mathematical logic. In this edition, the author has added an extensive appendix on second-order logic, a section on set theory with urelements, and a section on the logic that results when we allow models with empty domains. Introduction to Mathematical Logic