# Book on the Rigorous Foundations of Mathematics- Logic and Set Theory

I am asking for a book that develops the foundations of mathematics, up to the basic analysis (functions, real numbers etc.) in a very rigorous way, similar to Hilbert's program. Having read this question: " Where to begin with foundations of mathematics" I understand that this book must have:

• Propositional Logic
• First Order-Predicate Logic
• Set Theory

Logic however must not depend on set theory! I have tried reading many pdf notes on the first two but have been dissapointed by the usage of notions and concepts from Set Theory.

So what do I want? A book that builds up these $3$ from ground $0$ and develops the foundations of mathematics up to the Axioms of ZFC and simple consequences like the existence of the real number field. As such, it is not neccessary for this book to contain the incompleteness theorems, cardinality etc. It must however be rigorous and formal in the sense that when I finish it, I have no doubt that the foundations are "solid".

Final notes: It would be preferable if it were made for self study (but that's not neccessary). You can also suggest up to 3 books that discuss the topics above, beware however as circular definitions must be avoided. Rigor in other words, is the most important thing I am asking for.

PS: There have been other questions here on the foundations of logic as this one. They do not answer my question however, as rigor is not (over)emphasised. I believe this is not a duplicate and I hope you see that as well.

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I don't think it's possible for you to have no doubt that the foundations are solid without simply convincing yourself of it, because foundations really can't be shown to be solid, not without a significant paradigm shift, due to Godel's incompleteness theorems. –  tomasz Dec 25 '12 at 11:24
I can't say I don't understand your sentiment, but I think you're overrating formalism. Still, it's a matter of philosophy and not what this site is for. Good luck, anyhow. :) –  tomasz Dec 25 '12 at 11:45
Note that it isn't rigor you're asking for, but foundationalism. Beware infinite regress -- you have to accept something before you can even get started. And for the purposes of actually doing mathematics, circular definitions are required. Although I prefer the visualization of spiral definitions: when we use set theory to construct a theory of formal logic which we use to define set theory, we really need to distinguish between the two versions of set theory, lest we fall pray to various paradoxes (e.g. Skolem's) –  Hurkyl Dec 25 '12 at 11:46
@Doug: "Constructing" may have been a better word than "doing". If nothing else, whatever prior notion you accept to get started (e.g. a prior notion of manipulating strings of symbols) is eventually something you want to prove things about. But really, internalization of logic is the big issue I have in mind; i.e. using logic to prove things about sets and objects built from sets only really works well when you use a logic constructed within set theory, which is necessarily distinct from (but ideally similar to) the logic we used to define set theory to begin with. –  Hurkyl Dec 25 '12 at 13:51
@Nameless: I like Terence Tao's answer on this MathOverflow thread, as well as Tom Goodwillie's comment: "If you're looking for utter certainty, then even mathematics is not entirely the right field." I'd also mention my answer here (not that I am claiming it is on par with either of the answers I mentioned already). –  Zev Chonoles Dec 25 '12 at 14:01

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!

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Do you have any constructive contribution to suggest to answer the question? Or is this post an elaborate critique of other answers (comment) which may be too long to have posted as a comment? At best, it suggests what NOT to read. –  amWhy Feb 18 '13 at 17:15
Well, indeed, that was too long for a comment. I wasn't/am not clear enough about what the OP wanted to offer a positive suggestion (though perhaps clear enough that Bourbaki is very unlikely to soothe whatever foundational worries were prompting the question). –  Peter Smith Feb 18 '13 at 17:25
Fair enough! I just came across the post again as there is a new post that linked this one in a comment: perhaps you can help with that new post. –  amWhy Feb 18 '13 at 17:28

Unless I've missed something, both J. Lukasiewicz's Elements of Mathematical Logic and A. N. Prior's Formal Logic, start things from logic without any set theory required. Lukasiewicz's book takes you up to first-order predicate logic, while Prior's book takes you up to set theory, though I haven't read that far personally in the book, and will only get you started there. You might also want to look at the metamath site.

I do want to remark here that oftentimes rigor isn't oftentimes even ("over")emphasized in books written by logicians. Even demonstrations of something like "p implies p" given in sections on propositional logic are NOT usually formal proofs.... which might lead one to argue that p$\implies$p is true, but not provable. Fortunately, the above references generally don't have this problem (though Prior's section on "the logic of classes" (set theory) does), and the problem isn't, in principle, all that difficult to fix.

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I will look them up. Thank you –  Nameless Dec 25 '12 at 15:17

I think the best approximation of the book you are looking for is the series of books by Nicolaus Bourbaki. In the introduction to his books you can find the following (or similar:) statement: "this series of books takes up mathematics at the beggining and gives complete proofs". You can't find anything that is closer to Hilbert's program than Bourbaki's treatise.

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Thank you very much –  Nameless Dec 25 '12 at 15:15

Principia Mathematica was an attempt in 1910. Bourbaki was published 20 years later and has a completely different approach in terms of logic. Bourbaki does not deal with logic. The standard way this works is, you write a book on set theory, throw in one or two references about "paradoxes" and then proceed with mathematical metaphysics (Cantor's paradise if you will). If you really want to read a rigorous book, I would suggest Frege's concept script, Russell's theory of denoting phrases, Whitehead's universal algebra, Wittgenstein's tractatus from 1918. So be aware that the doctrines which are being served to you, by the standard channels, including this site and mathoverflow is only one part of the actual history. If you read people like Descartes, Leibniz, Poincare, Brouwer, Wittgenstein you will get a completely different view, which is almost a complete mirror of the tradition of Hilbert. Brouwer for example vanished from the history. "Working mathematicians" for the most part don't take intuitionism or discussions about foundations seriously.

By the way, Gödel used the aforementioned PM as the basic system, and Bourbaki is not such a system at all. Bourbaki starts with sets. Other systems are for example VonNeuman's system, Quine's new foundation, Tarski-Grothendieck system, Lawvere's category theory system, etc.

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So Bourbaki is not the right choice. But out of all the manuscripts (should I call them that?) that you have posted, which one deals with (classical) logic and ZF(C)? I know Principia and New Foundation don't for example. And where can I find them? –  Nameless Dec 26 '12 at 7:14
This is a vast subject and in my opinion Bourbaki and mathematical logic is purely formal and there is little to be learned by this exercise. Hilbert's program was put to rest by Gödel more or less. I would start here: en.wikipedia.org/wiki/Introduction_to_Mathematical_Philosophy and here plato.stanford.edu/entries/logical-atomism, and here plato.stanford.edu/entries/frege, and here en.wikipedia.org/wiki/Automated_theorem_proving. Read a book on elementary logic by Quine for example and compare with the questions asked by Russell, Frege. –  RParadox Dec 26 '12 at 10:25
Very nice links. I will read them as well as elementary logic and maybe one day write my own book to answer this question. Thank you. –  Nameless Dec 26 '12 at 10:28
Checkout metamath and state of the art Automated proving. There are some minor proofs given by computers. This goes back to AI, because if a computer can do mathematics based on axioms, why not program Hilbert's program in a computer. Hofstadter tried to answer this in Gödel, Escher, Bach. Wittgenstein gave a different view on logical atomism. None of this is in the standard curriculum of mathematics, because that would go against the prevailing paradigm. And so everything is build around a certain view. –  RParadox Dec 26 '12 at 10:28
For a newer discussion check out: New Directions in the Philosophy of Mathematics: An Anthology (Revised and Expanded Edition) by Thomas Tymoczko –  RParadox Dec 26 '12 at 11:19