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More Rigorous Text than Rudin?

  1. Nov 2, 2013 #1
    I was recommended Rudin's "Principles of Mathematical Analysis" as a text that assumes you know nothing and takes it from there. And make no mistake, I think it's an amazing book. I've learned techniques and overcome some hurdles all on my own that make me feel quite good about myself. So this is not a criticism of Rudin's work.

    I just feel that it's not rigorous enough for my taste. While starting out, I tried to imagine that Rudin wasn't talking about "numbers" when using terms like "-x" and "0" etc, but this became impossible later on.

    For example, while constructing real numbers from the rationals using Dedekind cuts, he suddenly talks about the "Archimedean" property of the set of rationals Q which he hasn't mentioned before. In fact, he seems to take the set of rationals for granted entirely without defining what Integers are, what natural numbers are, and indeed what "numbers" are in the first place.

    He also doesn't define a "set" or how they're constructed. Now I know a lot of this since I've worked with analysis before, but for my own satisfaction I want to start with a completely blank slate. Basically assume that I have infinite intelligence (a rash proposition!) but know absolutely nothing.

    Can anyone help me out with a text that starts from scratch...absolutely from nothing and then builds up to the construction of the real numbers? Like I said, I love Rudin and plan to continue studying the book but I also feel "incomplete" without having a rigorous understanding of some of the fundamentals that Rudin seems to take for granted.

    Any suggestions would be greatly appreciated!
     
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  3. Nov 2, 2013 #2

    UltrafastPED

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  4. Nov 2, 2013 #3
    Correct me if I'm wrong, but isn't that considered somewhat dated by today's standards?
     
  5. Nov 2, 2013 #4

    UltrafastPED

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    So? You asked for something that starts at the beginning ...
     
  6. Nov 2, 2013 #5

    jgens

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    Honestly I think this request is seriously misguided. To save you some cash, however, there is a construction of all these things outlined below:
    1. Take 0 = { }, 1 = {0}, 2 = {0,1}, ... , n = {0,...,n-1}, ... and this gives us our set of natural numbers. Order these guys by set membership and it is easily checked that this set is well-ordered. Defining addition is easy and we say that n+m is the unique natural number order isomorphic to the disjoint union n ⊔ m when endowed with the obvious order. Multiplication can now be handled recursively as repeated addition. By the way everything has been defined we get commutativity and associativity and distributivity automatically so we have constructed N.
    2. To get the integers start with the cartesian product NxN and define an equivalence class by taking (n1,m1) ~ (n2,m2) if and only if n1+m2 = n2+m1. It is easy to show that every equivalence class has a representative of the form (n,0) or (0,n). So take Z to be these equivalence classes and let the positive integers be those with representative (n,0) and the negative integers be those with representative (0,n). Addition and multiplication and the ordering are easy enough to define now since these properties are inherited from the natural numbers.
    3. To get the rationals start with the cartesian product Zx(Z-{0}) and define an equivalence relation where (n1,m1) ~ (n2,m2) if and only if n1m2 = n2m1. We let Q the equivalence classes of this relation and again the addition and multiplication and ordering are inherited from Z in a natural way.
    If you are honestly interested in these constructions, then can fill in the missing details above. They are all pretty trivial and unenlightening. If you want to understand these number systems, then the best way to do that is not by examining their construction, but by examining the consequences of the properties those number systems have.
     
  7. Nov 2, 2013 #6
    Well, going from N to Z to Q is really an algebra thing. Analysis takes over when you go from Q to R (Cauchy completeness/Dedekind cuts). The construction of N, Z, Q are not that difficult. They are fairly straightforward.

    Natural numbers are well, natural. There's not much to talk about. You can talk about defining a successor function. Google knows that. Going from N to Z is just adding in additive inverses (an algebraic thing). Going from Z to Q is just adding in multiplicative inverses (another algebraic thing).

    A book on analysis wouldn't cover algebra, as it is not an algebra text. What you're looking for is probably some more algebra. I recommend Dummit and Foote.
     
  8. Nov 2, 2013 #7
    Also "start from scratch". I've thought about books like that. But to truly start from scratch, it's pure logic and philosophy. It would take countless volumes of books to even begin to talk about sets or any math at all. I don't even know if there is a "beginning". Because you can get into logic/philosophy and just keep going back.

    Basically every book has to start from SOMEWHERE. Every book has to assume some knowledge, some axioms, some something. Not to mention there's controversy over certain philosophical questions surrounding math. Different approaches, etc.
     
  9. Nov 2, 2013 #8

    UltrafastPED

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    That's why I suggested "Principia Mathematica"!
     
  10. Nov 2, 2013 #9
    It's a horribly outdated book. I wouldn't recommend anybody to read it. There are much better books out there which don't make things as complicated, and which do things the modern way.

    I honestly don't know why people even remember the principia. Its relevance and influence on modern math is close to zero.

    Anyway, if you want to start at the beginning, but if you still want a rather comfortable book, get Hrbacek and Jech's book on set theory: https://www.amazon.com/Introduction-Edition-Revised-Expanded-Mathematics/dp/0824779150

    It's not a book on analysis though. For analysis, I doubt there are books which do books as deeply as you want, but check "foundations of real and abstract analysis" by Bridges, or "Real Analysis" by McShane and Botts.
     
  11. Nov 2, 2013 #10
    Thank you. Yours is the first actual suggestion :). I appreciate it.
     
  12. Nov 2, 2013 #11
    There's some good stuff here, but there's a lot I don't understand. What is "well ordered", what is "isomorphic", what is an "equivalence class"? So far, Rudin hasn't addressed these concepts in his book and I don't know if he will do so later on. Besides, since he already assumes we know about natural numbers it wouldn't be of much use anyway.
     
  13. Nov 2, 2013 #12
    If you don't know those then the issue is not a lack of rigor by Rudin but that you do not have sufficient background for the book. R136 gave a good suggestion with Jech.

    You may as also be interested in an algebra book. Virtually any undergraduate level algebra book would cover the essentials of set theory that one uses in analysis. Further, an isomorphism is an algebraic concept (in this context).
     
    Last edited: Nov 2, 2013
  14. Nov 2, 2013 #13
    I'm not sure about the background part. In the preface, he says "This book is intended to serve as a text for the course in analysis that is usually taken by advanced undergraduates or by first year students who study mathematics". It would seem to me to be the starting point since I completed my undergraduation with a course in Math.

    Besides, it's not as if I don't understand what "Archimedean" is for example. It's just that I haven't seen those concepts proved formally and that was what I was looking for. Are you sure that everyone who reads Rudin knows about Peano's axioms for example? In short does everyone who studies Rudin know everything about the number theory of natural numbers?
     
  15. Nov 2, 2013 #14

    WannabeNewton

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    A typical freshman year honors calculus class would have covered all of that (save for detailed properties of the natural/real numbers-something like the Archimedean property would have definitely been covered as it is a very elementary result) and real analysis a la Rudin would have been the next step. It's a waste of time to go through things like the construction of the naturals in a real analysis book when such things are done in texts on more foundational aspects of math (i.e. set theory). It would be much more pragmatic to take such constructions for granted and focus more on something like the construction of the reals, via for example the Cantor construction, because this is actually pertinent to real analysis.
     
  16. Nov 2, 2013 #15

    AlephZero

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    That doesn't really matter. The only practical way to learn math is to start somewhere in the middle, not at "the beginning". You started learning the useful properties of integers and rational numbers right back in elementary school, before you could even define what a number was (except for vague ideas like "it's something you use to count things").

    If you want to digress from real analysis into investigating the completeness and consistency of various axiom systems for mathematics, thats fine, just so long as you remember that isn't what real analysis is about.

    IMO the main problem with Russell and Whitehead is not so much that it "starts at the beginning", but that R&W were trying to figure out where the beginning actually was. With the benefit of almost 100 years of hindsight, it's possible to write a more coherent (and shorter) version. Reading the introductions to the sections is an interesting eyewitness account of history being made, though.

    There are better ways to learn calculus than to first learn Latin (or German) and then read the first editions of Newton and Liebniz. The same applies to R&W.
     
  17. Nov 2, 2013 #16

    WannabeNewton

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    ^Indeed, well put! It's like going through a differential topology text and expecting the author to develop the entire theory of integrals starting from measure spaces and working down to the Riemann integral before delving into integration of differential forms.
     
  18. Nov 2, 2013 #17
    Thank you - that was a very insightful comment. Perhaps what I'm really looking for after all is not real analysis after all, but the study of the foundations of mathematics. I just picked up Rudin because it was recommended to me as a text that assumes you know nothing, so I was speaking from that perspective.
     
  19. Nov 3, 2013 #18

    jgens

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    I certainly knew nothing about PA when I first read Rudin, but that hardly impacted my ability to understand what was going on in the book. The book provides a set of properties that characterize the real numbers uniquely up to isomorphism and this is all you need.

    No one knows close to everything about the theory of natural numbers. There are still questions about N that are active topics of research. The one thing I can guarantee you though, is that understanding properties of N is not going to come from understanding one particular construction of it.
     
  20. Nov 3, 2013 #19

    atyy

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    There is no beginning. An analogy from Cori and Pelletier's "Mathenmatical Logic" I like is that it's a spiral. You always begin on the Nth level and there are levels above and levels below.
     
  21. Nov 4, 2013 #20
    is the Hubbard Analysis a good text? Rudin proves everything in the slickest way possible, which maybe matters to some people
     
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