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Math Help - Set-theory challenge

  1. #1
    Senior Member Dinkydoe's Avatar
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    Set-theory challenge

    This is a question I was asked on my oral exam for set-theory. I found it a particular interesting one:

    Show that:

    For any infinite set X, there exists a bijection f:X\times X \to X \Leftrightarrow well-orderings-theorem of Zermelo.


    It's in particular the implication \Longrightarrow that's interesting.

    Hint 1.
    Spoiler:
    For \Longrightarrow : Make a smart choice for X


    Addition hint 1.
    Spoiler:
    Let A an arbitrary infinite set. Choose: X= A\cup \aleph(A)
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Dinkydoe View Post
    This is a question I was asked on my oral exam for set-theory. I found it a particular interesting one:

    Show that:

    For any infinite set X, there exists a bijection f:X\times X \to X \Leftrightarrow well-orderings-theorem of Zermelo.


    It's in particular the implication \Longrightarrow that's interesting.

    Hint 1.
    Spoiler:
    For \Longrightarrow : Make a smart choice for X


    Addition hint 1.
    Spoiler:
    Let A an arbitrary infinite set. Choose: X= A\cup \aleph(A)
    Instead of well-ordering can we use Zorn's or AOC?
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  3. #3
    Senior Member Dinkydoe's Avatar
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    Well in fact that's allowd too! If you show:

    for all infinite X there exists a bijection f: X\times X\to X\Leftrightarrow AOC, Lemma of Zorn, Wellorderings Theorem of Zermelo.

    They're all equivalent ofcourse.
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