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Math Help - Ordinal Problem.

  1. #1
    Junior Member
    Joined
    Apr 2008
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    48

    Ordinal Problem.

    Hi, the question posed to me is as follows:

    Prove or find a counterexample to:

    1. \alpha\cup\{\alpha \} is an ordinal \Rightarrow \alpha is an ordinal
    2. \alpha\cup\bigcup{\alpha} is an ordinal \Longleftrightarrow \alpha is an ordinal

    I've found proofs to both of these questions but I'm not entirely sure that they are both correct. Can anyone see a way these implications do not hold?

    Thank you.
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  2. #2
    Senior Member
    Joined
    Feb 2010
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    Thanks
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    a u {a} is an ordinal <-> a is an ordinal.

    a is an ordinal -> a u Ua is an ordinal.

    But it it does not hold for all a that

    a u Ua is an ordinal -> a is an ordinal.

    You should find a simple example of an a such that a u Ua is an ordinal but a is not an ordinal.
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