  We expand the notion of characteristic formula to infinite finitely
presentable subdirectly irreducible algebras. We prove that there is a
continuum of varieties of Heyting algebras containing infinite finitely
presentable subdirectly irreducible algebras. Moreover, we prove that there is
a continuum of intermediate logics that can be axiomatized by characteristic
formulas of infinite algebras while they are not axiomatizable by standard
Jankov formulas. We give the examples of intermediate logics that are not
axiomatizable by characteristic formulas of infinite algebras. Also, using the
Goedel-McKinsey-Tarski translation we extend these results to the varieties of
interior algebras and normal extensions of S4
