So, I recently started learning about modal logic, and thus also Possible Worlds semantics.
The formulation for a frame I was given is:
A frame is defined as an ordered pair $[G,R]$ where $G$ is the non-empty set of possible worlds, and $R$ is a relation $R \subseteq G \times G$.
So, it's obvious that this formulation uses set theoretic notions, but is this necessary? Are there formulations of modal logic that don't require set theory?
Modal logic is an attempt to axiomatize the intuitive idea of modal thought, where statements are true in some universe and false in others. The set theory definition of "frame" is just a rigorous set theory version of that idea.
However, the formal syntax of Modal Logic does not require set theory, any more than the formal syntax of Peano arithmetic does not require set theory.
Rather, a "frame" is just a starting idea before you get to the formal syntax, so you can understand the "semantics" of an expression in modal logic: "What does this expression 'mean?'"
Often, we construct models in set theory or category theory or number theory, to see the possible consistency or inconsistency of a logical system. Set theory is the most common, because it is more intuitive than category theory, and more powerful than number theory.