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Math Help - Ordinals

  1. #1
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    Ordinals

    Prove that if X is a nonempty set of ordinals, then \bigcap X is an ordinal. Moreover, \bigcap X is the least element of X.

    Does anyone have any suggestions?
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  2. #2
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    Quote Originally Posted by chimp View Post
    Prove that if X is a nonempty set of ordinals, then \bigcap X is an ordinal. Moreover, \bigcap X is the least element of X.

    Does anyone have any suggestions?
    [Trichotomy of ordinals]
    Let \alpha, \beta, \gamma be ordinals. Then, it satisfies one of the alternatives, \alpha \in \beta, \alpha=\beta, \beta \in \alpha.

    Lemma 1. Let \alpha be an ordinal. Then, any member of \alpha is itself an ordinal number.

    If \bigcap X = \emptyset, we are done. It is an ordinal number. Otherwise, it is a member of some ordinal number \gamma in X. By lemma 1, it is an ordinal number.

    We show that x = \bigcap X is the least element of X. If x = \emptyset, we are done.

    If x \neq \emptyset, we claim that x is the least element (w.r.t \epsilon-image) in X.
    Take any y \in X. If  x \in y, we are done again. Otherwise, if x \notin y, then y \in x or y=x by trichotomy of ordinals. This implies that y \subseteq x. Since y is an element of X, x \subseteq y. This forces y=x. Thus x is the least element of X.
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