Some Brouwerian Counterexamples Regarding Nominal Sets in Constructive Set Theory

The existence of least finite support is used throughout the subject of nominal sets. In this paper we give some Brouwerian counterexamples showing that constructively, least finite support does not always exist and in fact can be quite badly behaved. On this basis we reinforce the point that when working constructively with nominal sets the use of least finite support should be avoided. Moreover our examples suggest that this problem can't be fixed by requiring nominal sets to have least finite support by definition or by using the notion of subfinite instead of finite.