Within a fairly weak formal theory of numbers and number-theoretic se- quences we give a direct proof of the contrapositive of countable finite choice for decid- able predicates. Our proof is at the same time a proof of a stronger form of it. In that way we think that we improve a proof given by Diener and Schuster. Within the same theory we prove properties of inhabited sets of naturals satisfying the general contra- positive of countable choice. Extending our base theory with the continuity principle, we prove that each such set is finite. In that way we generalize a result of Veldman, who proved, actually within the same extension, the finiteness of these sets, supposing additionally their decidability.