Question regarding initial segment and transfinite induction.

Question

Suppose $X$ is an uncountable well-ordered set with $\leq$.For $x\in X$ ,define the initial segment of $X$ determined by $x$ as $I(x)=\{y\in X| y\leq x $ and $y\neq x\}$.Now my question is ,does there exist an $x\in X$ such that $I(x)$ is countably infinite?I think intuitively that it should occur but higher cardinalities are not clear enough for me.Can someone guide me a bit?

Answer 1

Yes, there are many such $x$: the first is $\min\{y\in X:I(y)\text{ is not finite}\}$. This question and answer point the way to showing that it is only countably infinite.