Finite Frames Fail: How Infinity Works its Way into the Semantics of Admissibility

Many intermediate logics, even extremely well-behaved ones such as
IPC, lack the finite model property for admissible rules. We give conditions
under which this failure holds. We show that frames which validate all admissible
rules necessarily satisfy a certain closure condition, and we prove
that this condition, in the finite case, ensures that the frame is of width 2.
Finally, we indicate how this result is related to some classical results on
finite, free Heyting algebras.