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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.

Thank you in advance

Community
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Nameless
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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
  • @tomasz By solid I mean as solid as can be. I don't care about the possible inconsistency of ZFC right now. – Nameless Dec 25 '12 at 11:26
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    Then what do you care about, exactly? To establish mere existence of real number field you don't need all that much set theory, and I think naive set theory is good enough for that. The most cumbersome step is, I think, taking the Dedekind-completion of rationals. But that's not really that much of a stretch. – tomasz Dec 25 '12 at 11:40
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    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
  • But the naive part which we use to formalize the basic parts of logic is easily integrated into ZFC. It suffices to learn basics of logic using naive set theory, and the basics of axiomatic set theory. – Asaf Karagila Dec 25 '12 at 11:45
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    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) –  Dec 25 '12 at 11:46
  • @AsafKaragila I would prefer is this naive part is reduced to a minimum but I understand your point. – Nameless Dec 25 '12 at 11:46
  • @Hurkyl Fundationalism! I didn't know such word existed! Well, of course you must accept some basic staments, but I would prefer if these were reduced to a minimum as I said before. Are you sure circular definitions are required? – Nameless Dec 25 '12 at 11:49
  • @Nameless: I want to make use of formal logic to study set theory. Therefore, set theory must be constructed by formal logic. In my set theoretic universe, I want to talk about groups and do non-standard analysis. This requires a formal logic constructed within the universe. –  Dec 25 '12 at 11:58
  • @Hurkyl I fail to understand why that's necessary (i.e. for formal logic to be constructed from within the universe (which I assume you mean set theory)). But if it is, then my question has no answer... – Nameless Dec 25 '12 at 12:00
  • @Hurkyl I simply do not see how circular definitions are "required" for doing mathematics. – Doug Spoonwood Dec 25 '12 at 13:14
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    @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. –  Dec 25 '12 at 13:51
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    @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
  • @Nameless: Some of the suggestions in the math StackExchange thread Where to begin with foundations of mathematics may be of help to you. – Dave L. Renfro Dec 27 '12 at 20:18
  • @DaveL.Renfro As you can see in my question, I was already aware of this. But thanks anyway – Nameless Dec 27 '12 at 21:11
  • @Nameless: Oops! Somehow I totally missed that, although I have no idea how. Maybe because a lot of things started happening at work when I went to look up that earlier post, which kept me from looking very carefully at your post. – Dave L. Renfro Dec 27 '12 at 22:49
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    @tomasz, you write: "foundations really can't be shown to be solid... due to Godel's incompleteness theorems." Well I agree with the first part, but its a mistake to bring incompleteness into the mix. To prove that a system $S_0$ is solid, I need to use another system $S_1$. This second system $S_1$ will necessarily have rules and axioms. How to make sure they're solid? Well I could invent another system $S_2$ that verifies that $S_1$ is solid. But this process never ends! And here's my point; this process never ends irrespective of Goedel's theorems.... – goblin GONE Jun 12 '14 at 12:58
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    ... In particular, if we lived in some kind of crazy reality where Goedelian incompleteness simply didn't occur, nonetheless we'd still have the aforementioned infinite regress. So, I think it is a mistake to bring Goedel into this. Your observation that "foundations really can't be shown to be solid" is correct, but this is due to the nature of the axiomatic method, and really has nothing to do with incompleteness. – goblin GONE Jun 12 '14 at 13:00
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    @user18921: It's been a while, but I believe my point was that, if it wasn't for the incompleteness, we could imagine that this process can somehow be completed in finitely many steps. At some point, one of your $S_n$s could verify that $S_n$ is solid (without being inconsistent), and then you would have no infinite regress. Whether that is enough for a solid foundation is debatable, but without it, the best we can get is the infinite regress. – tomasz Jun 12 '14 at 18:45
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    @tomasz, I understand what you're saying, but I think its worth noting that even without the incompleteness theorem, we still have a problem. Maybe some $S_n$ proves itself to be both sound and complete, but why should we trust $S_n$? The process terminates with $S_n$, but only God knows this, so for us "mere mortals", the process has not terminated. I guess at heart we're both agreeing, just emphasizing different aspects of the problem. – goblin GONE Jun 12 '14 at 18:53
  • Did you find what you searched? – Marco Medina Jul 01 '21 at 21:26
  • If you want absolute rigor, nothing can be better than a formalized system. Mentioned several times here, but metamath and coq are good candidates. So what's a formalized system: axioms, statements and proofs are coded into computer; each step of argument is checked by the computer using steps that anyone can follow. Such a way guarantees that there is no error and any chance of making hand-wavy arguments. – Student Jan 11 '22 at 19:42
  • Indeed the foundations are not solid at all. There is no foundation at all after all, just an illusion of foundation. – Masacroso Jan 19 '24 at 21:41

6 Answers6

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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!

Peter Smith
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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
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Curry's Foundations of Mathematical Logic is very conscious of what is presupposed in terms of mathematical content in the development of logic. The writings of Paul Lorenzen might also be of some interest for you.

sro
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    Perhaps this is exactly what I wanted. Can you add more information as to what Curry's FML and Paul Lorenzen's writings contain? Thanks – Nameless Jan 03 '13 at 15:48
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I precisely had this purpose of building the foundations of mathematics (logic and set theory) from ground 0 with absolute rigor (I mean, as much rigor as actually possible) with my site settheory.net.

There I start by building set theory and logic in parallel. I only break the chain of maximum rigor in the developments of model theory in Part 3, where I introduce and explain things more intuitively, not so rigorously. However I also have plans to bring rigorous foundations for these things, through the part on Galois connections, that you can see done and stands as a logical continuation of Part 2. There as you can see, I reached the concept of well-founded relation. This concept can be used to rigorously define the structure of formal expressions with their interpretations.

user27887
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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.

Doug Spoonwood
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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.

Godot
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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.

RParadox
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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 07:14
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    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: http://en.wikipedia.org/wiki/Introduction_to_Mathematical_Philosophy and here http://plato.stanford.edu/entries/logical-atomism/, and here http://plato.stanford.edu/entries/frege/, and here http://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