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Foundation of analysis proof

  1. Dec 16, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi everybody, i'm going through the book Fondations of Analysis by E. Landau. I'm trying to prove theorems by myself and then checking if they are correct. But i proved this theorem in a different way then the book and i need a check. thank you :wink:

    Theorem 4
    To every pair of numbers x,y we may assign an unique number x+y such that

    1) for every x x+1=x'

    2) for every x and y x+y'=(x+y)'

    2. Relevant equations
    Axiom 2

    for each x there exist an unique number called the successor of x, denoted by x'

    Axiom 4

    if x'=y' then x=y

    3. The attempt at a solution
    I proved existence in the same way of the book and so i know it's right.
    For uniqueness i did this.

    Let's take x,y to be arbitrary natural numbers.
    We assume there there exist z and w such that x+y = z and x+y = w and that they satisfies properties 1) and 2).
    By axiom 2

    (x+y)' = z and (x+y)' = w'

    Then by property 2)
    x+y' = (x+y)' = z'
    = w'

    and by axiom 4

    z = w

    The book prove this by induction, so maybe i missed something.:biggrin:
  2. jcsd
  3. Dec 16, 2012 #2
    Nevermind i see where it's wrong :grumpy:
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