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Thread: a fun proof of inf integers, tell me what you think.

  1. #1
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    a fun proof of inf integers, tell me what you think.

    Thought of this today.
    two years ago someone told me that you cannot know if there exist a biggest integer or not.
    then i said that there must be a bigger integer, because we can add one to it.

    But now i thought $\displaystyle \infty+1 = \infty$, so when we get to $\displaystyle \infty$ big numbers adding one does not increase it.

    keep in mind i have a very elementary knowedge of infinity, so maby this is supid i dont know.

    In this proof i will assume that:
    1. No integer can be both odd and even at the same time.
    2. For any integer n, $\displaystyle n+1\not< n.$

    1 and 2 are not proven here, but proof exist.

    assertion: there exist $\displaystyle \infty$ many integers.
    proof: suppose n is the biggest integer such that $\displaystyle n = n+1.$ But if n is even, then from def $\displaystyle n = 2k,$ for an integer k.

    But then $\displaystyle n+1 = 2k+1,$ for some k, which is odd by def. Then we know that $\displaystyle n\neq n+1,$ because of assumption 1. And we know that $\displaystyle n+1 \not< n$ by assumption 2.
    Then the only option that remains is $\displaystyle n+1 > n,$ But then n is not the biggest integer.

    But if n is odd then from def $\displaystyle n = 2k+1,$ for some k, But then $\displaystyle n+1 = 2k+2 = 2(k+1),$ which is even an even integer from def. Thus by assumptions 1,2 the only option is $\displaystyle n+1 > n.$
    therefore if n is odd or even, there exist a bigger integer n+1. Q.E.D.


    please point out any flaws in my arguments you may find.
    thanks
    Last edited by engpro; Jan 26th 2013 at 09:38 AM.
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  2. #2
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    Re: a fun proof of inf integers, tell me what you think.

    Hey engpro.

    The proof looks OK but is there a specific reason why you wanted to show it this way as opposed to another way?
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  3. #3
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    Re: a fun proof of inf integers, tell me what you think.

    no not really, i am still a novice on proofs, so i thought that this would be simple and fun to try to prove.
    i am sure that there exist many better and simpler proofs of this fact, but this is what i came up with.
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