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Math Help - uncountable set

  1. #1
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    uncountable set

    Let S_\omega be the minimal uncountable well-ordered set.
    (a) Show that S_\omega has no largest element.
    (b) Show that for every \alpha \in S_\omega, the subset \{x | \alpha < x\} is uncountable.
    (c) Let X_0 be the subset of S_\omega consisting of all elements x such that x has no immediate predecessor. Show that X_0 is uncountable.

    I know how to do (a) and (b) but don''t know (c), can anyone help?
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    Let S_\omega be the minimal uncountable well-ordered set.
    (a) Show that S_\omega has no largest element.
    (b) Show that for every \alpha \in S_\omega, the subset \{x | \alpha < x\} is uncountable.
    (c) Let X_0 be the subset of S_\omega consisting of all elements x such that x has no immediate predecessor. Show that X_0 is uncountable.

    I know how to do (a) and (b) but don''t know (c), can anyone help?
    Claim: If Y is subset of S_\omega in which every element has an immediate predecessor it follows that Y is countable.

    Thus, given any \alpha\in X_0 the largest subset of S_\omega that contains \alpha and in which every element has an immediate predecessor is countable.
    It follows that if X_0 were countable, then S_\omega would be a countable union of countable sets, hence countable.
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