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

Undecidability of the problem of recognizing axiomatizations for implicative propositional calculi