Intuitionistic notions of boundedness in N

We consider notions of boundedness of subsets of the natural numbers N that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and formulate the closely related notion of a detachably …nite subset. We establish metric equivalents for a subset of N to be detachably …nite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable …niteness and sequential continuity. Most of the results do not require countable choice. 1 Pseudobounded subsets A subset A of the natural numbers N is bounded if A is contained in a …nite subset of N. In this paper we consider other notions of boundedness of subsets of N that occur when working within intuitionistic logic. The goal is to understand a notion introduced by Ishihara [3] who called a subset A of N pseudobounded if for any sequence an in A, the sequence an=n converges to zero. This seems like a peculiar de…nition, but we know from [3] that the concept has some interesting connections with sequential continuity of functions between metric spaces. The following theorem provides some equivalent conditions for a subset of N to be pseudobounded. Examples of the sequence sn in the theorem are n, n, and b p nc.