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Some Good Books for Math Undergrads

A Customer 2000
The list author says: "Here are some good books to checkout if you like Math at the Undergrad/Grad level (mostly undergrad).

I tried to pick books which were for the most part user-friendly (except for Rudin). I also only know of 1 differential equations book so I put it down (actually, there's one more I know of that Mathematicians think is good- which means it's not a good intro)."
A Mathematical Introduction to Logic, Second Edition
A Mathematical Introduction to Logic, Second Edition
"Best book ever for mathematical logic.  The exposition is wonderful.   His basic recursion theory is the best out there.  He's a VERY rare combination, someone who teachs it well and knows what he is doing."
Elements of Set Theory
Elements of Set Theory
"Great book by a great guy.  He is the master of written mathematical exposition."
Linear Algebra Done Right (Undergraduate Texts in Mathematics)
Linear Algebra Done Right (Undergraduate Texts in Mathematics)
"This is a great "determinant free" book for the most part.  It's on the theory side so maybe not for someone who hasn't had an intro. "computational" linear algebra course."
Contemporary Abstract Algebra Fifth Edition
Contemporary Abstract Algebra Fifth Edition
"Easy to understand.  Good start."
Abstract Algebra, 3rd Edition
Abstract Algebra, 3rd Edition
"This is a comprehensive (if there is such a thing nowadays) and "easy-to-understand" basic algebra book for beginning grad and curious undergrads."
Rings, Modules and Linear Algebra (Mathematics Series)
Rings, Modules and Linear Algebra (Mathematics Series)
"I thought this book was out of print, but if this is what I am thinking of, it should be a great introduction to "Rings, Modules, and Linear Algebra."  Very easy to follow (especially on things such as Jordan canonical form)."
A First Course in Probability (8th Edition)
A First Course in Probability (8th Edition)
"This book is great once you've had an intro. course.  It's good for people starting upper div. probability (and for people thinking about being actuaries and taking the P1 exam).  Ross has tons of examples of varying difficulty."
Introduction to Abstract Algebra
Introduction to Abstract Algebra
"Great for self-study.  Has a lot of answers."
Algebra (2nd Edition) (Featured Titles for Abstract Algebra)
Algebra (2nd Edition) (Featured Titles for Abstract Algebra)
"If you've had a solid course in linear algebra (or similar course in understanding proofs), this book is a great introduction to algebra.  It's a classic."
Introduction to Analysis (Dover Books on Mathematics)
Introduction to Analysis (Dover Books on Mathematics)
"Great exposition and CHEAP since it's Dover."
Model Theory : An Introduction (Graduate Texts in Mathematics)
Model Theory : An Introduction (Graduate Texts in Mathematics)
"This is a good (modern) treatment of Modal Theory for beginning grad students or curious undergrads."
Notes on Set Theory (Undergraduate Texts in Mathematics)
Notes on Set Theory (Undergraduate Texts in Mathematics)
"Great book to look at after Enderton's.  Moschovakis writes well."
A Shorter Model Theory
A Shorter Model Theory
"I like this book.  People seem to like the more modern book by Marker (which is superb).  I'm a bit old-fashioned and still like Chang and Kiesler as well."
Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (International Series in Pure & Applied Mathematics)
Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (International Series in Pure & Applied Mathematics)
"I included this book because it's a classic.  But, I don't know how well you will learn from it.  The proofs are very slick.  If you're looking for a read where you don't put effort in, then this is not the book for you."
Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds
Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds
"This book is great, but not necessarily a starting book (depends on what you like I guess).  Most people will prefer Stewart, Rogawski, Sallas, etc for standard multivariable calc.  After that and a little bit of linear algebra, this book would be fun."
Applied Combinatorics, Second Edition
Applied Combinatorics, Second Edition
"Great text.  It covers a wide variety of topics and applications.  Easy to understand."
A Course in Combinatorics
A Course in Combinatorics
"From the CalTech course.  Good book.  Chapters are more or less individual lectures."
Introduction to Topology: Second Edition (Dover Books on Mathematics)
Introduction to Topology: Second Edition (Dover Books on Mathematics)
"This book has good exercises (and answers).  It's exposition is clear and concise.  Greene is a great guy.  He always wants you to see the idea (not just the details).  Gamelin is excelent with mathematical prose for the non-mathematican (the mathematician as well)."
Topology (2nd Edition)
Topology (2nd Edition)
"A classic.  Skip the first chapter and get straight to the topology.  More comprehensive than Gamelin and Greene."
Complex Analysis (Undergraduate Texts in Mathematics)
Complex Analysis (Undergraduate Texts in Mathematics)
"Modern and clear.  There's also a solutions manual in progress (not by the author) somewhere on the web for public display.  I prefer this to Ahlfors (but then again some people would think I'm a moron for doing so)."
Elementary Differential Equations and Boundary Value Problems
Elementary Differential Equations and Boundary Value Problems
"The only diffEQ I know."
Algebra (Graduate Texts in Mathematics)
Algebra (Graduate Texts in Mathematics)
"Great reference text. Grad level."
Basic Algebra (Cornerstones)
Basic Algebra (Cornerstones)
"Another great reference text (grad level)."
Introduction to Smooth Manifolds (Graduate Texts in Mathematics)
Introduction to Smooth Manifolds (Graduate Texts in Mathematics)
"As far as grad books go, this is incredible."
Introduction to Topological Manifolds (Graduate Texts in Mathematics)
Introduction to Topological Manifolds (Graduate Texts in Mathematics)
"As far as grad books go, this book is king if you want to learn the material quickly.  I like this book much better than Hatcher, but, then again,  I never seriously used Hatcher (so maybe I'm wrong)."
Naive Set Theory (Undergraduate Texts in Mathematics)
Naive Set Theory (Undergraduate Texts in Mathematics)
"This is a very nice, well-written book by a great mathematics writer.  His linear algebra is also fantastic if you want a more "upper div." approach."
Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics)
Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics)
"Great book for a grad level text.  Less dense than Jech."
Introduction to Lie Algebras (Springer Undergraduate Mathematics Series)
Introduction to Lie Algebras (Springer Undergraduate Mathematics Series)
"I've always liked the Springer undergrad series.  I guess at this level, it's more about how you say it.  Most grad books seem to take the approach of "it's what you say" not "how you convey it."  If only Springer had an undergrad version of its grad series."
Real Analysis (4th Edition)
Real Analysis (4th Edition)
"I like this grad level book over Folland as an intro."