uncountable set

**dori1123** · September 15th 2009, 05:02 PM

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?

**Failure** · September 20th 2009, 09:45 AM

Originally Posted by dori1123

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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