Good books on mathematical logic? I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation?
 A: Shoenfield's "Mathematical Logic".  The notation is a bit dated, but the exercises are great.
A: Here is an opinionated and detailed 100 page Study Guide to logic textbooks, updated fairly regularly.
A: François G. Dorais and others made some great recommendations to me some time ago over on MathOverflow. They're fairly high-level (not exactly introductory courses) but they're good reads.
A: Cori, Lascar, Pelletier, Mathematical Logic: A course with exercises -- Part I and Part II. Especially the second one.
A: You should take a look at D. Van Dalen: Logic and Structure. 
A: I suggest to read:


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*The incompleteness phenomenon, by H. Judah and M. Goldstern.

*There is a very good on-line course notes by L. van den Dries: http://www.math.uiuc.edu/~vddries/410notes/main.dvi

*A Mathematical Introduction to Logic by H. Enderton.

*Mathematical logic, by H.-D. Ebbinghaus, J. Flum and W. Thomas.

A: For my work in this area, I refer to:

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*Richard Epstein "Classical Mathematical Logic"

*Wolfgang Rautenberg "A Concise Introduction to Mathematical Logic"

*Jon Barwise "Handbook of Mathematical Logic"

*Jean Heijenoort "From Frege to Gödel"

*Wei Li "Mathematical Logic: Foundations for Information Science"

Rautenberg has a lot of examples, exercise, but is very heavy going (at least for me).  Epstein is fairly recent and very well laid out. While, Barwise is the most comprehensive for when you need to deep dive.
A: Yuri Manin, A course in mathematical logic.
A: I recently started studying from An Introduction to Mathematical Logic and Type Theory: To Truthe through Proof by Peter B. Andrews. It's great at my level of mathematical knowledge. Perhaps this is more introductory than you are looking for.
A: The following web page includes my list of 46 low to medium price textbooks on mathematical logic from 1940 to 2004 in chronological order.
http://www.topology.org/tex/conc/logicbooks.html
There's such a wide variety of topics and approaches in logic, it's difficult to give a particular recommendation. So my web page contains lists of which books cover which topics, including a cross-reference table of which topics are in which books, in chronological order.
A: A book that should be read by everyone in mathematics regardless of level is Wolfe's A Tour Through Mathematical Logic. 
It's simply a compulsory read, I couldn't put it down. It gives a broad overview of mathematical logic and set theory along with its history, and it is absolutely beautifully written. That's the best place for anyone to begin. 
A: Ebbinghaus, Flum and Thomas. Mathematical Logic (Amazon)
A: I know this is late by more than a decade, but you could give "Introduction to Logic and the Methodology of Deductive Sciences" by Alfred Tarksi a try, although I haven't read the entire book, I found the first few chapters of part I very readable. Here are some links to reviews of the book:
https://www.projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-48/issue-7/Review--Alfred-Tarski-Introduction-to-Logic-and-to-the/bams/1183504438.full
https://www.goodreads.com/book/show/673488.Introduction_to_Logic
https://projecteuclid.org/journals/modern-logic/volume-8/issue-1-2/Review-of-Alfred-Tarski-Introduction-to-Logic-and-to-the/rml/1081878077.pdf
Happy reading!
A: I will give recommendations based on some stuff I actually tried reading.

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*Stephe Kole Kleene's Logic book

This is the most detailed logic book that I have ever read in my life. It will give answers to foundational questions that you could not even imagine. One problem I felt with it is that it doesn't do a very good job bridging the logic we use in Natural language to the forma language. This leads me to the next recommendation.


*Modern Logic : Graeme Forbes

This is the only book I know which tries to explain in detail how the ideas of Logic can be applied to see Natural language itself in a different way. I am sure that a person who has tried studying the Grammar of at least one Natural language would really appreciate this book.
