Pedersen: From Hölder to Hahn: Comparative, Classificatory, and Quantitative Conceptions in Probability and Decision

From Hölder to Hahn: Comparative, Classificatory, and Quantitative Conceptions in Probability and Decision
Arthur Paul Pedersen (LMU Munich)
4:10 pm, Friday, April 12th, 2019
Faculty House, Columbia University

Abstract. This talk is a contribution to measurement theory with a special focus on the foundations of probability and decision theory. Its primary goals are two-fold. One goal is to introduce an extension of a mathematical result itself generalizing measurement theory’s cornerstone, Hölder’s Theorem, which assures that any Archimedean totally ordered group is isomorphic to a subgroup of the additive group of real numbers. This extension, Hahn’s Embedding Theorem, drops the Archimedean requirement to obtain an isomorphism to a subgroup of a Hahn lexical vector space. The mathematical result I report in this talk establishes an isomorphism to an explicit, commensurate ordered field extension of the real numbers—numbers therein expressed in terms of formal power series in a single infinitesimal, with familiar additive and multiplicative operations equipped with a lexicographic order—measurement theory’s answer to admitting non-Archimedean systems without recourse to routine techniques from non-standard analysis.

The second goal of this talk is to explain what motivates all this math, turning to measurement theory’s paradigmatic applications to introduce this talk’s central result. Following a review of objectionable prohibitions and permissions attributed to standard probability and decision theories, such as conditions of dominance, countable additivity, continuity, and completeness, the second mathematical result I report establishes, as a corollary to the first, that reasonably regimented preferences are representable by subjective expected utility permitting non-Archimedean-valued probability and utility, as needed—and likewise for regimented comparative probability systems as well as for a faithful recasting of de Finetti’s coherent previsions. These corollaries, already free of typical structural requirements, extend to admit indeterminacy and incompleteness.

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