It follows constructively from weak versions of Markov's principle (MP) and of Church's thesis (WCT) that logical validity for single sentences is not arithmetically definable. This is a direct consequence of the constructive model-theoretic fact that, using MP and WCT, one can prove the categoricity of first-order Heyting arithmetic. By a straightforward refinement of this proof, we then obtain generalizations of the incompleteness theorem of Kreisel [K 2] as presented by van Dalen [Da] and improved by Leivant [L 1]. An analogous result for classical validity in r.e. models appears in [V].Our methods of proof are new in that they rely not upon variants of the ideas of [K2] but upon a constructive proof idea inspired by the classical result of Tennenbaum ([Te], [E&K], [S]) on recursive models of arithmetic. This line of reasoning was suggested by the applications of readability to recursive mathematics described in [McC1], [McC2] and [McC3].