Consider the following definition for the index set for a stochastic process
The set $T$ is called the index set or parameter set of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set $T$ the interpretation of time. In addition to these sets, the index set $T$ can be other linearly ordered sets or more general mathematical sets, such as the Cartesian plane $R^2$ or $R^n$, where an element $t \in T$ can represent a point in space. But in general more results and theorems are possible for stochastic processes when the index set is ordered.
It is clear from the the definition that it can be infinite. Can the index set be finite?
Yes, the index set can be any set at all. In general, a definition means nothing more or less than what it says, so if the definition calls $T$ just a set, then it can be any set.