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Mathematical logic- need some suggestion

  1. Nov 9, 2009 #1
    Mathematical logic-- need some suggestion

    I'm planning to start self-study of mathematical logic and axiomatic set theory. In fact I have already started and but facing a lot of problems to grasp the conception and formalism used there. After studying Hilbert's program and Godel 's Incompleteness theorem I was stunned and I thought I must learn formal math. I always had a fascination to know the foundation of math from 11th grade.
    Now being an engineering student I already have had subjects like advanced calculus, differential equation, complex variables and Fourier transform, probability and linear algebra. But they don't seem to help any more in mathematical logic. What I'm watching here is a bunch of alphabets, symbols and they are suffering me a lot. I can't understand them well.
    I've got the related books of mathematical logic by J. Donald Monk, Helmut Schwichtenberg, Michal Walicki, J. Adler, J. Schmid, total 4 books but they are tough. I never did any course on logic before and it seems to me that mathematical logic is really really a tough one.
    I have two inquiries,
    1. what are the prerequisites of mathematical logic such that I can understand these peculiar alphabets and symbols and underlying meanings of them?

    2.I didn't study topology yet. I intended to start topology after finishing mathematical logic. Is that a wise approach?
    I would really appreciate your help. Please also give me some advice about how to approach this type of hard subject. Thank you.
     
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  3. Nov 9, 2009 #2

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    Re: Mathematical logic-- need some suggestion

    Logic only requires the famous "mathematical maturity" as a background. It is really just about actually understanding precise definitions and making deductions from axioms. Something like abstract algebra might help you work more easily in that setting, but other than that, you really don't have a choice except studying the subject itself. I've read Cohen's Set Theory and the Continuum Hypothesis and Manin's A Course in Mathematical Logic. They are both good intros and I don't really know any better ones. Cohen's book gets tough at the end, but the beginning is easier and doesn't get bogged down in the notation and symbol pushing in a way many books on logic do.
     
  4. Nov 9, 2009 #3
    Re: Mathematical logic-- need some suggestion

    Yes, while I'm not fully fluent in set theory or mathematical logic (but certainly not unversed, either), but I can say that one of two courses will help you:

    1. Math from a mathematician's perspective
    2. Abstract algebra

    The first course varies its name from school to school. Basically, most universities (that I've seen) offer a course in teaching students how to actually think in terms of mathematical logic, teaching them how to make acceptable proofs from fundamental axioms, and teaching them how to think in terms of abstract objects in place of concrete ones (this allows you to think analytically, critically, and in generalities for mathematical purposes).

    Your best bet, if none of these seem appealing, is to take an "Introduction to Logic" in your university's philosophy department. They should deal with the necessary propositional and predicate logic that you'd need to understand the mathematical versions of them (for which there are only minor differences).
     
  5. Nov 9, 2009 #4
    Re: Mathematical logic-- need some suggestion

    My favorite is Ebbinghaus and Flum's Mathematical Logic. It will take you through to the incompleteness theorems and formal undecidability as well as some additional topics. You will probably want to have at least a course in Abstract Algebra nonetheless.

    Donald Monk's book is a terrible, terrible place to start. Don't pick it back up until you have mastered another book (and preferably have learned a little theory of recursion), and then it is probably valuable mostly as a reference and an alternative approach to the subject.

    That being said, if you haven't taken an intro to higher mathematics class or some other class that focuses on theorem proving, you are going to be lost. Even after you take a course it might be a while before you have good intuition for what is being done. The mathematical logic texts look the following: The mechanics of what is being done in theorem proving, The mechanics of mathematical structures in general such as 'The theory of Groups' or 'Number Theory' or 'Ring/Ideal Theory' ect, and also look at how far you can take formalization via the theorems of Godel.

    An interesting result of looking at the general mechanics of mathematical structures is a field called 'Non-standard analysis' which is essentially Integral and differential calculus(the rigorous form of which is called Real Analysis) but it allows for infinitely small and infinitely large numbers so that you can avoid messy epsilon delta notation and look at real analysis in a different light. The key thing about Non-standard analysis is that most anything (not quite everything) that can be proven in it holds true in normal Real Analysis.

    As far as Set theory. I would recommend Set Theory the third millennium edition by T. Jech. If that is too heavy, go for Introduction to Set theory by Jech and Hrbacek. Personally I think that the first one is nicer and if you can get through it you will be up to speed on current research areas, plus it's the one I'm working on right now =p.

    As far as a sub-area of mathematical logic, you could look into Wilfrid Hodges' Model Theory after you've had a couple courses in abstract algebra. It's pretty neat from what I've read so far.
     
  6. Nov 9, 2009 #5
    Re: Mathematical logic-- need some suggestion

    First and foremost, if you want to take a glimpse at the foundation of modern math I suggest you take a look at Howard Eves' book, and also the following: http://www.abstractmath.org/" [Broken]

    I don't suggest you learn mathematical logic or axiomatic set theory before you seriously understand what the above have to say. You really can't study those two subjects without an already firm foundation, or at least, you won't understand their significance if you don't.
     
    Last edited by a moderator: May 4, 2017
  7. Nov 10, 2009 #6
    Re: Mathematical logic-- need some suggestion

    Thank you all of you for your suggestions. At least I have now some clues from your suggestions about the approach to study mathematical logic. Thanks again
     
  8. Nov 10, 2009 #7
    Re: Mathematical logic-- need some suggestion

    You may also want to take a look at things from the more philosophical side. Godel, Russel, etc were philosophers, and philosophy talks more about the foundations of the subject. See http://plato.stanford.edu/entries/philosophy-mathematics/ for a basic summary and be sure to check out the related entries at the bottom, especially those more related to logic.

    If it's an option, see if you can get into the "Advanced Logic" course in your philosophy department. "Symbolic Logic" may be a prerequisite.
     
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