Fred Richman

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Is it reasonable to do constructive mathematics without the axiom of countable choice? Serious schools of constructive mathematics all assume it one way or another, but the arguments for it are not compelling. The fundamental theorem of algebra will serve as an example of where countable choice comes into play and how to proceed in its absence. Along the(More)
I was inspired, not to say provoked, to write this note by Michel J. Blais's article A pragmatic analysis of mathematical realism and intuitionism [2]. Having spent the greater part of my career doing intuitionistic mathematics, while continuing to do classical mathematics, I have come to feel that most comparisons of these two approaches to mathematics(More)
A common procedure for selecting a particular density from a given class of densities is to choose one with maximum entropy. The problem addressed here is this. Let S be a …nite set and let B be a belief function on 2 . Then B induces a density on 2 , which in turn induces a host of densities on S. Provide an algorithm for choosing from this host of(More)
A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete archimedean Heyting …eld, a terminal object in the category of archimedean Heyting …elds.
We present group-theoretic and cryptographic properties of a generalization of the traditional discrete logarithm problem from cyclic to arbitrary finite groups. Questions related to properties which contribute to cryptographic security are investigated, such as distributional, coverage and complexity properties. We show that the distribution of elements in(More)
At the center of the theory of abelian p-groups are the classical theorems of Ulm, Zippin and Kaplansky, going back to the thirties, that classify countable p-groups by their Ulm invariants: the uniqueness theorem is referred to as Ulm's theorem, the existence theorem as Zippin's theorem. For each ordinal , the -th Ulm invariant of G can be de ned as the(More)