formula manipulation, I tried to find the simplest formula $\phi(x)$ with one free variable $x$ in the language of ZFC that defines the first transfinite ordinal $\omega$ (i.e. $\phi(x) \leftrightarrow x=\omega$). Here is my attempt (operator precedence follows the Wikipedia ): $$\begin{align}\phi(x):=\exists a (a \in x) \land \forall a \forall b