What are the ordinals that can be defined by a first order formula?

Question

When I read my textbook of logic and set theory, I often see formulae that are use symbols that do not belong to the language. For example, $\emptyset$ is not part of the language, but we can use it in a formula because we know how to translate the latter into a real formula of the language. This makes me wonder what are the ordinals that we can just use in a formula as a shorthand notation? More precisely, what are the ordianls $\alpha$ for which there exists a first order formula $F[v_0]$ such that $\alpha$ is the only element in the universe that satisfy $\mathcal{U}\models F[\alpha]$? I know how to define standard integers in this way. For example, $0=\emptyset$ can be defined by $F[v_0]=\forall v_1 \neg v_1\in v_0$, and $1$ can be defined by $G[v_0]=\forall v_1 v_1\in v_0 \Leftrightarrow (\forall F[v_1] \vee \forall v_2\;( v_2\in v_1\Leftrightarrow F[v_2] )) $, etc. We can also define $\omega$ in this way because we can express "$v_0$ is the smallest limit ordinal" with a formula. I was going to say that we cannot define every ordinal in this way because there are uncountably many ordinals but only countably many formulae. But then, there are also countable models of ZFC. So what can we say?