Is there a first-order sentence defining a cardinal?

Question

Being an ordinal is first-order definable, within ZFC at least.

Out of many possible first-order characterizations, one is that an ordinal is a transitive set where $\in$ is trichotomous, i.e. $w$ is an ordinal if and only if:

1. $[\forall x]([\exists a](x \in a \land a \in w) \to x \in w)$
2. $[\forall x \forall y](x \in y \lor x = y \lor x \ni y)$

Being a Reinhardt cardinal is not first-order definable.

However, the much more basic question of whether being a cardinal is first-order definable doesn't seem to exist yet.

Is being a cardinal first-order definable?

Answer 1

Yes, here's one way to do it. It's kind of unsatisfying because it isn't succinct like the first-order characterization of an ordinal.

As has been established, being an ordinal is a first-order property.

Two sets having the same cardinality as each other also has a first-order characterization. $A$ and $B$ have the same cardinality if and only if there exists a bijection between them and a bijection is a set of pairs satisfying some additional first-order condtions.

A cardinal is the least ordinal of a given cardinality. Thus $w$ is a cardinal if and only if:

1. $w$ is an ordinal.
2. For every ordinal $v$ of the same cardinality as $w$, it holds that $v = w$ or $w \in v$.

These two conditions can be expanded into a long first-order formula.