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Thread: uncountable set

  1. #1
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    uncountable set

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

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

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