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Divergent mathematical treatments in utility theory
In this paper I study how divergent mathematical treatments affect mathematical modelling, with a special focus on utility theory. In particular I examine recent work on the ranking of information states and the discounting of future utilities, in order to show how, by replacing the standard analytical treatment of the models involved with one based on the framework of Nonstandard Analysis, diametrically opposite results are obtained. In both cases, the choice between the standard and nonstandard treatment amounts to a selection of set-theoretical parameters that cannot be made on purely empirical grounds. The analysis of this phenomenon gives rise to a simple logical account of the relativity of impossibility theorems in economic theory, which concludes the paper
Vop\v{e}nka's Alternative Set Theory in the Mathematical Canon of the 20th Century: Author's Translation from Czech
Vop\v{e}nka's Alternative Set Theory can be viewed both as an evolution and
as a revolution: it is based on his previous experience with nonstandard
universes, inspired by Skolem's construction of a nonstandard model of
arithmetic, and its inception has been explicitly mentioned as an attempt to
axiomatize Robinson's Nonstandard Analysis. Vop\v{e}nka preferred working in an
axiomatic theory to investigating its individual models; he also viewed other
areas of nonclassical mathematics through this prism. This article is a
contribution to the mapping of the mathematical neighbourhood of the
Alternative Set Theory, and at the same time, it submits a challenge to analyze
in more detail the genesis and structure of the philosophical links that
eventually influenced the Alternative Set Theory.Comment: This is the author's translation into English of her paper published
originally in Czech. 14 page
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