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ARE integers ordered pairs of natural numbers:

  1. Aug 24, 2009 #1
    some rant about definition and semantics.

    integers are isomorphic ordered pairs of natural numbers (a,b) w/ equivalence relation (a,b)=(c,d) iff a+d=b+c.

    reals are convergent sequences of rationals,


    in mathematics, are integers simply isomorphic to the ordered pairs of natural numbers w/ the equivalence relation? or is the set of integers equal to the set of pairs of natural numbers with the equivalence relation?

    is it really correct to say that "the set of naturals is a subset of the set of reals" or "reals are a subset of complexes"?
    Last edited: Aug 24, 2009
  2. jcsd
  3. Aug 24, 2009 #2
    It doesn't matter in the slightest. You can define them however you want (as long as you end up with an isomorphic structure).
  4. Aug 24, 2009 #3
    It's all relative - reading a good book on the set theoretic foundations of numbers and mathematics in general will help clear this up.

    From the beginning

    From sets to natural numbers

    {} is called 0
    {{}, } is called 1
    {{{}, }, } is called 2, etc..

    From naturals to integers
    [0, 1] = [1, 2] = [2, 3] is called -1
    [0, 0] = [1, 1] = ... is called 0
    [2, 5] = [1, 4] = ... is called -3, etc.

    From integers to rationals
    (2, 1) = (4, 2) = ... is called 2
    (1, 1) = (2, 2) = ... is called 1
    (1, 2) = (2, 4) = ... is called 1/2

    From rationals to irrationals
    This is the sticky one and hinges around Cauchy sequences or Dedekind cuts.

    Since -3 is just a label and means different things in different contexts, but they are all in some sense the same thing (via isomorphism).

  5. Aug 24, 2009 #4
    and i would like to know is it really correct to say that "the natural numbers are a subset of the rationals", etc?

    i don't think so, if rationals are ordered pairs of ordered pairs of naturals, and naturals are just naturals.

    sorry if i sound rude, but how can you just say that 3 is a rational, and {{3,0},{1,0}} is a natural? it is just being sloppy, isn't it.

    edit:Here was the dirty secret. those mathematicians/textbooks were actually talking about isomorphisms!
    Last edited: Aug 24, 2009
  6. Aug 24, 2009 #5
    The number 3 is

    A Cauchy sequence converging to 3 AND it is
    An equivalence class of the ordered pair (3, 1) (as in = 3/1) AND it is
    An equivalence class of the ordered pair [3, 0] (as in 3 - 0) AND it is
    {{{{}, }, }, }

    So in reality it is a Cauchy sequence of ordered pairs of ordered pairs of successive inclusions of the empty set.

    Complex numbers would just be ordered pairs of such.

    Everthing can be boiled down to the brace notation. The label "3" is just a convenient glyph that represent this concept. Understandably the common sense of "3" predates the more contrived definition, but an axiomatic development of number requires this sort of bootstrapping from basic principles.

    Check out the draft by Ali Nesin:

    http://www.math.bilgi.edu.tr/courses/Lecture%20Notes/SetTheoryLectureNotes.pdf [Broken]

    Specifically chapters 10 through 13.

    Last edited by a moderator: May 4, 2017
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