Ask for recommendations for textbook on mathematical logic

I studied mathematical logic using a book not written in English. I would now like to study it again using a textbook in English. But I hope I can read a text that is similar to the one I used before, so I ask here for recommendations. Any recommendation will be appreciated. The characters of the mathematical logic book I used before is as follows:

1, Formal.

Everything is formal, introduced from very beginning, such as (I'm not clear if my expression is correct in English, that's why I need an English book to remedy this deficiency) basic bricks of mathematical logic (symbol, formulas, predicate word, logic word, constrained variable, etc.), the formation rules of propositional/predicate logic (how well-formed formula is constructed recursively), differnt systems of propositional/predicate logic such as P, P*. F, F* and their relation, formal reasoning rules in these systems(e.g., in P, there are five formal reasoning rules: $(\in),(\tau),(\neg),(\to_-) and (\to_+)$).

2, comprehensive

In addition to the conceptions in 1, It introduces also Sheffer vertical, function word and equality word. calculus of propositional/predicate logic including replacement of equal formula, the replacement theorem, normal forms, Skolem normal forms, Gentzen normal forms, non-embedding normal forms, reduction of logic calculus. Reliability. Assignment. Completeness of propositional/predicate (with equality word) logic. Compactness theorem and Lowenheim-Skolem theorem. Independency.

3, many examples

in this book, there are many examples of calculus of propositional/predicate logic using the formal reasoning rules of a specified system. It also introduces actual formal mathematical systems such as elementary algebra, natural numbers, definition of formal symbols, etc. It used a "diagonal" form of proof to prove things.

I hope to have a book in English similar to this one, that is, possesses these charactors. Could you please recommendate one? It would be better if the recommendated textbook can also contain some motivations and some modern things(the book I read was written 30 years ago). Thanks!

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Some standard suggestions may be found here: mathoverflow.net/questions/44620/undergraduate-logic-textbook –  Timothy Chow Apr 15 '11 at 14:02
This is a sufficiently international forum that it could possibly be helpful for you to name the original book. –  Alexander Woo Apr 15 '11 at 19:25
@Alexander Woo: The original book is "数理逻辑基础"(basics of mathematical logic), written by 胡世华(Hu Shihua) and 陆钟万(Lu zhongwan), published by Science Press, China. The first author, who is a academician, was said to be a son of a premier of former Beiyang government of China. –  zzzhhh Apr 16 '11 at 7:56

I was going to recommend the English translation of the two volume sequence by Cori and Lascar. But after reading again your message it is highly possible that this is the text you used. I really like these two introductory books.

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The construction rule for proposition formulas -- Definition 1.2 -- is the same as that of P* in the book I read before. It would be nice if this definition is followed by the eleven formal reasoning rules ($(\in),(\tau),(\neg),(\wedge_-),(\wedge_+),(\vee_-),(\vee_+),(\to_-),(\to_+),(\‌​leftrightarrow_-) \rm{and} (\leftrightarrow_+)$), and then by examples such as $A\to B|--|\neg A\vee B$ and DeMorgen laws that not only deliver useful theorems of reasoning frequently used in each branch of math but also illustrate the usage of these formal reasoning rules. (to be continued) –  zzzhhh Apr 16 '11 at 11:30
I'm surprised to find that no book introduces these formal reasoning rules, not to mention examples. After all, this is the most similar book so far. –  zzzhhh Apr 16 '11 at 11:30

For really learning how to do mathematics in a formal logic, I suggest to look at one of the theorem provers and read their manual or tutorial. For instance, you can take a look at HOL-light.

My experience is that books about logic, fall short if it comes to the art of really doing mathematics in logic. You asked about examples. If you take a book about logic, you will probably not find an example of a proof of a well known theorem in a formal logic. However, you will find it in the manuals of the theorem provers.

For me, some things got a place, when I studied HOL-light.

Regards,

Lucas

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This is really an interesting thing. I'll download it and make a study of it. Thank you. –  zzzhhh Apr 16 '11 at 8:44

I would suggest "A concise introduction to mathematical logic" by Rautenberg, Wolfgang. See Springer website at http://www.springer.com/mathematics/book/978-1-4419-1220-6 . I think it satisfies most of your requirements.

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Boolos and Jeffrey's book Computability and Logic may be of interest. In some respects its aim is breadth rather than depth. For example, the chapter on forcing does just enough to prove one interesting theorem (in arithmetic, not in set theory), and similar things are true of other topics. It seems to be addressed to mathematicians who don't know the topic and who want to find out what is of interest in the subject, not to people who want to learn all the skills needed to work in that area.

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The Better introduction I know is P. Johnstone "Notes on Logic and Set Theory" (Cambridge Mathematical Textbooks)

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Maybe it's not exactly what you had in mind (well, it was written about 30 years ago as well), but a nice book too.

H.D. Ebbinghaus, J. Flum, W. Thomas, Mathematical logic (Undergraduate texts in mathematics). Springer, 1984. http://books.google.com/books?id=VYLA8m7cqYcC

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