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Introduction to mathematical logic

My main reason for recommending Elliot Mendelson's Introduction to mathematical logic to someone wishing to study mathematical logic is the start of the third chapter (of 5). This chapter is on formal number theory, and starts off with a list of axioms and then some proofs. That's what mathematical logic is all about isn't it? Well maybe it's more about metaproofs - that is proofs about what you can prove, and certainly that is the main content of this book. However, I feel that having a few pages of the proofs you are dealing with is vital to give the student a foothold in this difficult subject, but, somewhat surprisingly, its difficult to find such proofs in books at this level.

The first chapter is on the (relatively simple) propositional calculus. The second deals with quantification theory - I would recommend that you look at the start of chapter 3 to start with, so as to have a concrete example. The later parts of chapter 3 get on to more complicated mathematical logic, such as Gödel's incompleteness theorem. Chapter 4 is on set theory. Now you may have come across set theory at school, but I have to tell you that the axiomatic version is a totally different ball game. But with the introduction to axiomatics from the previous chapters, this book would be a good place to start studying it. The fifth chapter is on computability. I wouldn't suggest this book as a starting point for this - it's much easier to approach it via real computers. However, it could be useful for seeing the links with other forms of logic.

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Hardcover 456 pages
ISBN: 0412808307
Salesrank: 678204
Weight:1.68 lbs

Published: 1997 Chapman & Hall/CRC

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Hardcover 456 pages
ISBN: 0412808307
Salesrank: 536568
Weight:1.68 lbs

Published: 1997 Chapman & Hall

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Hardcover 456 pages
ISBN: 0412808307
Salesrank: 258472
Weight:1.68 lbs

Published: 1997 Chapman & Hall/CRC

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Product Description
The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in mathematical logic. This edition includes an extensive appendix on second-order logic, a section on set theory with urlements, and a section on the logic that results when we allow models with empty domains. The text contains numerous exercises and an appendix furnishes answers to many of them.Introduction to Mathematical Logic includes:opropositional logicofirst-order logicofirst-order number theory and the incompleteness and undecidability theorems of Gödel, Rosser, Church, and Tarskioaxiomatic set theoryotheory of computabilityThe study of mathematical logic, axiomatic set theory, and computability theory provides an understanding of the fundamental assumptions and proof techniques that form basis of mathematics. Logic and computability theory have also become indispensable tools in theoretical computer science, including artificial intelligence. Introduction to Mathematical Logic covers these topics in a clear, reader-friendly style that will be valued by anyone working in computer science as well as lecturers and researchers in mathematics, philosophy, and related fields.

Product Description

The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in mathematical logic. This edition includes an extensive appendix on second-order logic, a section on set theory with urlements, and a section on the logic that results when we allow models with empty domains. The text contains numerous exercises and an appendix furnishes answers to many of them.Introduction to Mathematical Logic includes:opropositional logicofirst-order logicofirst-order number theory and the incompleteness and undecidability theorems of Gödel, Rosser, Church, and Tarskioaxiomatic set theoryotheory of computabilityThe study of mathematical logic, axiomatic set theory, and computability theory provides an understanding of the fundamental assumptions and proof techniques that form basis of mathematics. Logic and computability theory have also become indispensable tools in theoretical computer science, including artificial intelligence. Introduction to Mathematical Logic covers these topics in a clear, reader-friendly style that will be valued by anyone working in computer science as well as lecturers and researchers in mathematics, philosophy, and related fields.

A big mistake *
Late in August, the text originally selected for my mathematical logic class became unavailable. On the basis of reviews only, I chose Mendelson's Introduction to Mathematical Logic as the replacement. A disasterous choice. There may be a page without a typo, but I don't expect to find it. The presentation is inconsistent in notation and focus. Concepts are confused and more difficult than they should be. Definitions are not wisely selected. This book reads like something that has been patched for four decades (since 1964). On the positive side it contains interesting supporting material and will be a valuable private source of ideas to the lecturer. Be sure to read sections from chapter 2 and 3 before selecting this as a text.

A big mistake *

Late in August, the text originally selected for my mathematical logic class became unavailable. On the basis of reviews only, I chose Mendelson's Introduction to Mathematical Logic as the replacement. A disasterous choice. There may be a page without a typo, but I don't expect to find it. The presentation is inconsistent in notation and focus. Concepts are confused and more difficult than they should be. Definitions are not wisely selected. This book reads like something that has been patched for four decades (since 1964). On the positive side it contains interesting supporting material and will be a valuable private source of ideas to the lecturer. Be sure to read sections from chapter 2 and 3 before selecting this as a text.

A must have.... ****
This is a very useful and must have book for every graduate student in logic.Theory covers many fields(logic and computability) and has a lot of exercises (and also solutions to the tough ones)!!!

A must have.... ****

This is a very useful and must have book for every graduate student in logic.Theory covers many fields(logic and computability) and has a lot of exercises (and also solutions to the tough ones)!!!

twisted pants unleashed on men *****
This is one of the more popular introductory textbooks on mathematical logic, with Enderton's being its biggest competitor. I prefer Mendelson's for its breadth of material and the choice of proofs he uses, which are generally the most intuitive (e.g. Kalmar's for the completeness of the propositional calculus). This is not to say that they are always constructive, as they many of them are in the older texts (e.g. Kleene, Introduction to Metamathemaitcs). The exercises are thoughtfully chosen. There's a good range of difficulty and a good portion of the answers can be found in the back. Difficult questions are indicated to the reader. Out of all the mathematical logic texts I have (which are quite a few in number), this is the most oft-referred-to.

twisted pants unleashed on men *****

This is one of the more popular introductory textbooks on mathematical logic, with Enderton's being its biggest competitor. I prefer Mendelson's for its breadth of material and the choice of proofs he uses, which are generally the most intuitive (e.g. Kalmar's for the completeness of the propositional calculus). This is not to say that they are always constructive, as they many of them are in the older texts (e.g. Kleene, Introduction to Metamathemaitcs).

The exercises are thoughtfully chosen. There's a good range of difficulty and a good portion of the answers can be found in the back. Difficult questions are indicated to the reader.

Out of all the mathematical logic texts I have (which are quite a few in number), this is the most oft-referred-to.

Wonderful at the second glance. *****
Mendelson's Introduction to Mathematical Logic was the textbook for a logic-course I took a couple of years ago. At the time I did not like the book at all. It seemed too difficult and so typographically ugly that I thought I would never use it. Things have changed though. Now, I keep it close at hand on my desk and use it almost every day. Technical questions that used to require a trip to the library and several different books to answer, can usually be resolved by a look in Mendelson's book. It's wonderfully rich and clear! I still don't find everything easy but that's because the material isn't easy and so not something Mendelson can be blamed for. I do find the typography ugly and at times annoying, but that's a small price to pay for a presentation as rigorous and detailed as Mendelson's. So in summary: it's not the ideal book for the complete newcomer, but once you get past the initial hurdle it's a must read.

Wonderful at the second glance. *****

Mendelson's Introduction to Mathematical Logic was the textbook for a logic-course I took a couple of years ago. At the time I did not like the book at all. It seemed too difficult and so typographically ugly that I thought I would never use it. Things have changed though. Now, I keep it close at hand on my desk and use it almost every day. Technical questions that used to require a trip to the library and several different books to answer, can usually be resolved by a look in Mendelson's book. It's wonderfully rich and clear! I still don't find everything easy but that's because the material isn't easy and so not something Mendelson can be blamed for. I do find the typography ugly and at times annoying, but that's a small price to pay for a presentation as rigorous and detailed as Mendelson's.
So in summary: it's not the ideal book for the complete newcomer, but once you get past the initial hurdle it's a must read.

Best reference in first step math logic ****
Mendelson reaches an optimal point between the concision of the expert reference, and the wideness requested to a introductory text. Not in vain it has been the text forced in the universities during forty years. Nevertheless, I believe to have found an error in the demonstration that does of the theorem of the completeness of the Predicate calculus, in the part in which it tries to demonstrate that all logical truth is a theorem of the system. [...]

Best reference in first step math logic ****

Mendelson reaches an optimal point between the concision of the expert reference, and the wideness requested to a introductory text. Not in vain it has been the text forced in the universities during forty years.
Nevertheless, I believe to have found an error in the demonstration that does of the theorem of the completeness of the Predicate calculus, in the part in which it tries to demonstrate that all logical truth is
a theorem of the system.
[...]

Wonderful at the second glance *****
Mendelson's Introduction to Mathematical Logic was the textbook for a logic-course I took a couple of years ago. At the time I did not like the book at all. It seemed too difficult and so typographically ugly that I thought I would never use it. Things have changed though. Now, I keep it close at hand on my desk and refer to it frequently. Technical questions that used to require a trip to the library and several different books to answer, can usually be resolved by a look in Mendelson's book. It's wonderfully rich and clear! I still don't find everything easy but that's because the material isn't easy and so not something Mendelson can be blamed for. I do find the typography ugly and at times annoying, but that's a small price to pay for a presentation as rigorous and detailed as Mendelson's. So, in summary: it's not the ideal book for the complete newcomer (unless he or she comes to it with some mathematical sophistication), but once you get past the initial hurdle it's a must read. It's a little on the expensive side, but if you're serious about logic it is definitely worth it.

Wonderful at the second glance *****

Mendelson's Introduction to Mathematical Logic was the textbook for a logic-course I took a couple of years ago. At the time I did not like the book at all. It seemed too difficult and so typographically ugly that I thought I would never use it.

Things have changed though.
Now, I keep it close at hand on my desk and refer to it frequently. Technical questions that used to require a trip to the library and several different books to answer, can usually be resolved by a look in Mendelson's book. It's wonderfully rich and clear! I still don't find everything easy but that's because the material isn't easy and so not something Mendelson can be blamed for. I do find the typography ugly and at times annoying, but that's a small price to pay for a presentation as rigorous and detailed as Mendelson's.

So, in summary: it's not the ideal book for the complete newcomer (unless he or she comes to it with some mathematical sophistication), but once you get past the initial hurdle it's a must read. It's a little on the expensive side, but if you're serious about logic it is definitely worth it.

A Classic Textbook Now In Its Fourth Edition *****
Nearly forty years after it was published (1964), Elliot Mendelson's Introduction To Mathematical Logic still remains the best textbook on the principal topics of this subject. Although the book does not presuppose any background in the subject or in any particular branch of mathematics, the reader should have some degree of "mathematical sophistication." The first chapter starts with truth tables and ends with a completeness proof of a given formal system for propositional logic and an independence proof of the axioms of this system. Chapter Two is the study of quantification theory. Topics include quantificational completeness, Hilbert's Second Epsilon-Theorem, various topics from model theory, such as compactness and Lowenheim-Skolem Theorems, theorems on submodels and ultrafilters and non-standard analysis. The new fourth edition adds a very nice section on interpretations of quantification theory that allow the empty domain. Chapter Three presents an axiom system for number theory, recursive functions and proves (among other theorems) the famous Godel Incompleteness theorems, Tarski's indefinability of Truth Theorem and Church's Undecidability Theorem. Chapter Four is devoted to elementary set theory. Topics include an axiom system for set theory, ordinal and cardinal numbers, the axiom of choice and regularity, and alternative axiom systems of set theory. The new fourth edition includes an axiom system with urelements, something rarely presented, and an interesting note on the historical application of such a system in the construction of the first independence proof of the axiom of choice. The fifth chapter is the study of computability. The chapter begins with the notion of an algorithm and Turing Machines and builds up to the Kleene-Mostowski Hierarchy. The new fourth edition concludes with an excellent appendix on second-order logic. I have used Mendelson's book to teach a one-semester course to advanced undergraduate and graduate students with great success. Such a course is centered on the first three chapters, omitting from Chapter Two anything beyond quantificational completeness. If time permits, I recommend either the rest of Chapter Two, the beginning of Chapter Five, or the appendix on second-order logic. Set theory, the content of Chapter Three, is usually offered as a separate course.

A Classic Textbook Now In Its Fourth Edition *****

Nearly forty years after it was published (1964), Elliot Mendelson's Introduction To Mathematical Logic still remains the best textbook on the principal topics of this subject. Although the book does not presuppose any background in the subject or in any particular branch of mathematics, the reader should have some degree of "mathematical sophistication."

The first chapter starts with truth tables and ends with a completeness proof of a given formal system for propositional logic and an independence proof of the axioms of this system. Chapter Two is the study of quantification theory. Topics include quantificational completeness, Hilbert's Second Epsilon-Theorem, various topics from model theory, such as compactness and Lowenheim-Skolem Theorems, theorems on submodels and ultrafilters and non-standard analysis. The new fourth edition adds a very nice section on interpretations of quantification theory that allow the empty domain. Chapter Three presents an axiom system for number theory, recursive functions and proves (among other theorems) the famous Godel Incompleteness theorems, Tarski's indefinability of Truth Theorem and Church's Undecidability Theorem. Chapter Four is devoted to elementary set theory. Topics include an axiom system for set theory, ordinal and cardinal numbers, the axiom of choice and regularity, and alternative axiom systems of set theory. The new fourth edition includes an axiom system with urelements, something rarely presented, and an interesting note on the historical application of such a system in the construction of the first independence proof of the axiom of choice. The fifth chapter is the study of computability. The chapter begins with the notion of an algorithm and Turing Machines and builds up to the Kleene-Mostowski Hierarchy. The new fourth edition concludes with an excellent appendix on second-order logic.

I have used Mendelson's book to teach a one-semester course to advanced undergraduate and graduate students with great success. Such a course is centered on the first three chapters, omitting from Chapter Two anything beyond quantificational completeness. If time permits, I recommend either the rest of Chapter Two, the beginning of Chapter Five, or the appendix on second-order logic. Set theory, the content of Chapter Three, is usually offered as a separate course.

Outstanding Organization and Clear Style *****
I was sufficiently fortunate to have taken Professor Emeritus Mendelson's famous logic course at Queens College, the City University of New York, just two semesters before his retirement. I was, and continue to be, astonished by Dr. Mendelson's precise yet easy style, and the beautifully efficient organization of the subjects. Everything from the expository prose to the system of notational conventions has been carefully thought through so as to make the book both very substantive and very readable. In my opinion, it's the best introduction to serious mathematical logic currently on the market, and thanks to the genius of its author, it is likely to remain so for a long time. The buyer will not be disappointed.

Outstanding Organization and Clear Style *****

I was sufficiently fortunate to have taken Professor Emeritus Mendelson's famous logic course at Queens College, the City University of New York, just two semesters before his retirement. I was, and continue to be, astonished by Dr. Mendelson's precise yet easy style, and the beautifully efficient organization of the subjects. Everything from the expository prose to the system of notational conventions has been carefully thought through so as to make the book both very substantive and very readable. In my opinion, it's the best introduction to serious mathematical logic currently on the market, and thanks to the genius of its author, it is likely to remain so for a long time. The buyer will not be disappointed.