Existence of Minimal Uncountable Well-Ordered Set

Theorem
There exists a set of countable ordinals.

That is, there exists an uncountable well-ordered set $\Omega$ with the property that every initial segment in $\Omega$ is countable.

Proof using choice
We use the ZFC axioms.

We can identify the set of natural numbers with elements of a minimal infinite successor set.

The axiom of infinity and the axiom of subsets imply such a set exists.

By the axiom of powers, there exists the power set $\mathcal P\left({\N}\right)$.

This set is uncountable.

Thus there exists at least one uncountably infinite set.

Let $X$ be some uncountably infinite set.

By the well-ordering theorem, $X$ can be well-ordered.

Let $\left({X,\preceq}\right)$ be some well-ordering.

Let $X_a$ denote the initial segments of $X$ determined by $a \in X$

Let $P(S)$ be propositional function:


 * $X$ is uncountable and $X_a$ is countable for every $a \in S$

If $P(X)$, then $X$ is the sought-after set of countable ordinals.

If $\neg P(X)$, then there is some element $a \in X$ such that $X_a$ is uncountable.

By the definition of a well-ordering, there is a smallest element $a_0 \in X$ such that $X_{a_0}$ is uncountable.

Then the segment $X_{a_0}$ is itself uncountable, but every initial segment in $X_{a_0}$ is countable.

Set $\Omega = X_{a_0}$.

Also see

 * Uniqueness of Set of Countable Ordinals