Logic and Analysis 

Abstract: In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class.

Abstract: I develop a notion of nonlinear stochastic integrals for hyperfinite Levy processes and use it to find exact formulas for expressions which are intuitively of the form $$\sum_{s=0}^t\phi(\omega,dl_{s},s)$$ and $$\prod_{s=0}^t\psi(\omega,dl_{s},s)$$ , where l is a Levy process. These formulas are then applied to geometric Levy processes, infinitesimal transformations of hyperfinite Levy processes, and to minimal martingale measures. Some of the central concepts and results are closely related to those found in S. Cohen’s work on stochastic calculus for processes with jumps on manifolds, and the paper may be regarded as a reworking of his ideas in a different setting and with totally different techniques.

Abstract: We prove that the zero-set of a C∞ function belonging to a noetherian differential ring M can be written as a finite union of C∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zero-dimensional regular zero-set of functions in M consists of finitely many points. These results hold not only for C∞ functions over the reals, but more generally for definable C∞ functions in a definably complete expansion of an ordered field. The class of definably complete expansions of ordered fields, whose basic properties are discussed in this paper, expands the class of real closed fields and includes o-minimal expansions of ordered fields. Finally, we provide examples of noetherian differential rings of C∞ functions over the reals, containing non-analytic functions.

Abstract: In this article we use our previous constructions (L. Brunjes, C. Serpe, Theory Appl. Categ. 14:357-398, 2005) to lay down some foundations for the appli- cation of A. Robinson's nonstandard methods to modern algebraic geometry. The main motivation is the search for another tool to transfer results from characteristic zero to positive characteristic and vice versa. We give applications to the resolution of singularities and weak factorization.