What set theory axioms do I need to believe in uncountable ordinals?

Solution 1:

As [1], [2], [3], and [4] may tell you, the axiom of choice is not needed to define $\omega_1$ (in particular [1] and [4]).

Two key axioms are the power set axiom and the replacement axiom schema. In [2] and [3] you can see why the axiom of power set is needed. It is consistent with $\sf ZF$ without the power set axiom that there are only countable ordinals. In particular the set of hereditarily countable sets satisfies that. In fact, it shows that without the power set axiom we cannot prove the existence of any uncountable sets.

But the replacement schema is also essential. We use it in order to map a certain subset of $\mathcal P(\omega\times\omega)$ onto ordinals, and we need the replacement schema to show that the result is a set. Indeed if we consider $\sf ZF$ without the replacement schema, then $V_{\omega+\omega}$ is a model of these axioms, and there are only countable ordinals in that model. It should be remarked that there may still be well-ordered of length $\omega_1$ in $V_{\omega+\omega}$ but the von Neumann ordinal, a transitive set ordered by $\in$ does not exist there. That is to say, it is consistent that the axiom of choice holds, and every set can be well-ordered, but the von Neumann ordinals don't exist beyond $\omega+\omega$. In such model we separate between ordinals as we think about today (von Neumann's definition), and as equivalence classes of order types (which are proper classes, of course).

Of course one uses the axiom of union all the time, as well extensionality. It should be remarked that regularity is not necessarily because we can always limit ourselves to the part of the well-founded sets, where it holds, at least if we have replacement.

The Links:

1. How do we know an $ \aleph_1 $ exists at all?
2. Uncountable ordinals without power set axiom
3. Does the definition of countable ordinals require the power set axiom?
4. No uncountable ordinals without the axiom of choice?