Strange property of Well-Ordering on an Uncountable Set

Solution 1:

There's nothing mysterious about this at all. Think of the following simpler case, where "finite" and "infinite" take the roles above held by "countable" and "uncountable" respectively:

Any infinite well-order $S$ has an infinite set of elements with finitely many predecessors.

Basically, $\omega$ is an infinite well-ordering in which each element is finite. In general, for any cardinal $\kappa$, every well-ordering $W$ of cardinality $\ge\kappa$ will have an initial segment $\hat{W}$ isomorphic to $\kappa$, and every element of this $\hat{W}$ will have $<\kappa$-many predecessors (since $\kappa$ is a cardinal).

Solution 2:

Any well ordering of an uncountable set has an initial segment order-isomorphic to $\omega_1$. There are uncountably many elements of $\omega_1$, but each of those elements has the property that it is greater than only countably many elements.

In other words, $\omega_1$ is the set of all countable ordinals.