I am looking for a highly rigorous book on mathematical logic that goes into great detail, even at the foundations. I'm looking for something that say, develops propositional and first-order logic through discussion of free semigroups and words, rigorously proves unique readability, etc. Does such a book exist? Even some of the more rigorous books I've encountered have handwaived around alot. Thanks in advance for any response.
[Math] Highly Rigorous Logic Book
book-recommendationlogicreference-request
Related Solutions
Gosh. I wonder if those recommending Bourbaki have actually ploughed through the volume on set theory, for example. For a sceptical assessment, see the distinguished set theorist Adrian Mathias's very incisive talk https://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf
Bourbaki really isn't a good source on logical foundations. Indeed, elsewhere, Mathias quotes from an interview with Pierre Cartier (an associate of the Bourbaki group) which reports him as admitting
'Bourbaki never seriously considered logic. Dieudonné himself was very vocal against logic'
-- Dieudonné being very much the main scribe for Bourbaki. And Leo Corry and others have pointed out that Bourbaki in their later volumes don't use the (in fact too weak) system they so laboriously set out in their Volume I.
Amusingly, Mathias has computed that (in the later editions of Bourbaki) the term in the official primitive notation defining the number 1 will have
2409875496393137472149767527877436912979508338752092897
symbols. It is indeed a nice question what possible cognitive gains in "security of foundations" in e.g. our belief that 1 + 1 = 2 can be gained by defining numbers in such a system!
There are two very different kinds of question here:
What is logic exactly? ... What is "proof"? What is "truth"?
All good questions. But famously they do not have sharp, determinate, clear, uncontentious answers. Indeed, they are characteristically philosophical questions (that fall into the purview of what is often called "philosophical logic").
Of course, a technical logic text will introduce e.g. a sharp, technical, notion of a proof-in-a-given-formal-system (the fine print can be significantly different in different texts). But what is the relation between (1) the everyday notion of mathematical proof and (2) various notions of proof-in-a-given-formal-system which aim to model mathematical proof? This is up for (philosophical) debate. Similarly for the notion of truth, and indeed for the notion of a logic.
A "rigorous logic text" is therefore not the best place, really, to look for the discussion of the philosophical questions here. For those questions are (as it were) standing back from details in those rigorous texts and asking more general, philosophical, questions about them.
Please recommend me a good precise logic textbook.
Still, if you do want pointers to formal logic textbooks then there are a lot of suggestions, at various levels, on various areas of logic, in the Guide you can find at http://www.logicmatters.net/tyl
Best Answer
Kleene's Introduction to Metamathematics and Kleene's Mathematical Logic are two books that are well worth looking at for what you want. They're a bit dated, but I don't think this would be much of a problem for what you want. For what it's worth, on many occasions I've seen references (in papers and books) to one or both of Kleene's books for some technical or obscure point that is often overlooked or omitted in other texts.
The following excerpt from pp. 23-24 of Introduction to Metamathematics is an example of what I mean by a "technical or obscure point that is often overlooked or omitted in other texts".
[proof omitted in this excerpt]
[proof omitted in this excerpt]
[proof omitted in this excerpt]