For a complexity measure , a set is -infinite if it contains a -decidable infinite subset. For a time-based , we prove that there is a recursive S s.t. both S and its complements, , are infinite but not -infinite. Lipton [6] states that the existence of a recursive set S s.t. neither S nor os polynomially infinite is not a purely logical consequence of theorems of Peano's Arithmetic PA. His proof uses a construction of an algorithm within a non-standard model of of Arithmetic, in which the existence of infinite descending chains in such models is overlooked. We give a proof of a stronger statement to the effect that the existence of a recursive set S s.t. neither S nor is linearly infinite is not a tautological consequence of all true assertions. We comment on other aspects of [6], and show that a tautological consequence of true assertions may not be equivalent (in PA, say) to a sentence. The three sections of this paper use techniques of Recursion Theory, Proof Theory and Model Theory, respectively.