Several model-theoretic properties have multiple equivalent definitions. For a basic example, a theory $T$ is stable if

* there exists $\kappa\ge\|T\|$ such that for any model $M\models T$ and $A\subseteq M$ of size $|A|\le\kappa$, there are at most $\kappa$ complete types over $A$, or equivalently,

* there do not exist a formula $\phi(\bar x,\bar y)$, a model $M\models T$, and tuples $\{\bar a_n:n\in\omega\}$ in $M$ such that
$$M\models\phi(\bar a_i,\bar a_j)\iff i<j$$
for all $i,j\in\omega$.

You can start developing stability theory taking either property as a definition, and eventually deriving the other as a theorem.