26
$\begingroup$

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

  • 5
    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.2012-12-25
  • 0
    @tomasz By solid I mean as solid as can be. I don't care about the possible inconsistency of ZFC right now.2012-12-25
  • 1
    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.2012-12-25
  • 2
    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. :)2012-12-25
  • 0
    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.2012-12-25
  • 7
    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)2012-12-25
  • 0
    @AsafKaragila I would prefer is this naive part is reduced to a minimum but I understand your point.2012-12-25
  • 0
    @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?2012-12-25
  • 0
    @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.2012-12-25
  • 0
    @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...2012-12-25
  • 0
    @Hurkyl I simply do not see how circular definitions are "required" for doing mathematics.2012-12-25
  • 3
    @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.2012-12-25
  • 2
    @Nameless: I like Terence Tao's answer on [this MathOverflow thread](http://mathoverflow.net/questions/29104/why-are-proofs-so-valuable-although-we-do-not-know-that-our-axiom-system-is-cons), 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](http://math.stackexchange.com/a/50614/264) (not that I am claiming it is on par with either of the answers I mentioned already).2012-12-25
  • 0
    @Nameless: Some of the suggestions in the math StackExchange thread [Where to begin with foundations of mathematics](http://math.stackexchange.com/questions/140681/where-to-begin-with-foundations-of-mathematics) may be of help to you.2012-12-27
  • 0
    @DaveL.Renfro As you can see in my question, I was already aware of this. But thanks anyway2012-12-27
  • 0
    @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.2012-12-27
  • 1
    @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....2014-06-12
  • 0
    ... 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.2014-06-12
  • 0
    @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.2014-06-12
  • 1
    @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.2014-06-12

6 Answers 6