  In this paper we consider propositional calculi, which are finitely
axiomatizable extensions of intuitionistic implicational propositional calculus
together with the rules of modus ponens and substitution. We give a proof of
undecidability of the following problem for these calculi: whether a given
finite set of propositional formulas constitutes an adequate axiom system for a
fixed propositional calculus. Moreover, we prove the same for the following
restriction of this problem: whether a given finite set of theorems of a fixed
propositional calculus derives all theorems of this calculus. The proof of
these results is based on a reduction of the undecidable halting problem for
the tag systems introduced by Post.
