We develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. A full transfer principle and a strong idealisation principle are obtained by using a sheaf-theoretic construction due to I. Moerdijk. The construction is, in a precise sense, a reduced power with variable filter structure. We avoid the nonconstructive standard part map by the use of nonstandard hulls. This leads to an infinitesimal analysis which includes nonconstructive theorems such as the Heine-Borel theorem, the Cauchy-Peano existence theorem (...) for ordinary differential equations and the exact intermediate-value theorem, while it at the same time provides constructive results for concrete statements. A nonstandard measure theory which is considerably simpler than that of Bishop and Cheng is developed within this context. (shrink)

We introduce the notion of Boolean measure algebra. It can be described shortly using some standard notations and terminology. If B is any Boolean algebra, let BN denote the algebra of sequences , xn B. Let us write pk BN the sequence such that pk = 1 if i K and Pk = 0 if k < i. If x B, denote by x* BN the constant sequence x* = . We define a Boolean measure algebra to be a Boolean (...) algebra B with an operation μ:BN → B such that μ = 0 and μ = x. Any Boolean measure algebra can be used to model non-principal ultrafilters in a suitable sense. Also, we can build effectively the initial Boolean measure algebra. This construction is related to the closed open Ramsey Theorem 193–198.). (shrink)