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Are Spivak's properties of numbers provable?

  1. Dec 21, 2013 #1
    I know I am jumping ahead, as I am still working through part 1 of Spivak's Calculus, and am absolutely not properly equipped to prove his properties (if they are indeed provable).

    However, as I am trying to work on my proofing skills, my interpretation thus far has been, that these 12 "rules" are givens...unprovable. If you accept these givens, everything else follows from them, provably. If you do not accept these givens, you reject the proofs drawn from them.

    When I get into the higher level maths one day in the future, will these become provable?

    The rules are: associative law for addition and multiplication, existence of an additive identitfy and inverse, commutative law for addition and multiplication, distributive law, existence of multiplicate inverse, existence of multiplicative identity, trichotomy law, closure under addition, closure under multiplication, least upper bound property

    No need to try to prove any of them to me as I likely am not in a position to understand, just curious.

    Also, is there a name for these properties of numbers? Clearly Spivak did not create these properties, so I am wondering what the popular name for these properties is.

    Thank you in advance.
     
  2. jcsd
  3. Dec 22, 2013 #2
    Those are axioms, meaning that there is no proof for them - see http://en.wikipedia.org/wiki/Axiom. In maths, and indeed all logic really, all statements are ultimately based on some sort of assumption. Mathematics consists of taking a set of assumptions and making more and more complicated inferences from them to see what sort of interesting properties the system created by your assumptions might have, this is the case whether or not the author makes it explicit that what he is doing is based on some prior set of assumptions. That is my view of it, at least.

    As for what to call that set of statements, I guess you could name them the axioms of real numbers under addition and multiplication.
     
  4. Dec 22, 2013 #3
    As Spivak mentions at the end of the section, those properties "are merely explicit statements of obvious, well-known properties of numbers". And that is true, but, as he says, that is difficult to prove because "what numbers are" is "rather vague".

    In elementary school and up, we learn how to operate numbers practically, and we learn through a huge number of calculations and problems that all those properties hold whenever we check for them. But we do not really know what numbers are, we are simply not given any precise theory until much later (or never at all for most people). One such theory is developed from the Dedekind-Peano axioms, which carefully define all those silly little details of working with numbers that we just take for granted usually. And in that theory the 12 properties of Spivak are provable.

    But, the Dedekind-Peano axioms are still that, axioms, themselves not provable, even if what they say seems "obviously correct".

    A more abstract view would take the 12 properties as axioms, in which case they do not have to be provable - or, rather, cannot be provable to begin with. That is OK from the purely theoretic point of view, but it is not practical because it does not give us anything we can really use to compute things.
     
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