Abstract
The Axiom of Choice has, since its conception, formed an important part of the framework within which much of mathematics is done. Not only has its application allowed the proof of results to which it is an apparently essential pre-requisite, but its availability has shaped the way in which the mathematics which it proves has been formulated. There is no need to carry along the way the impedimenta of accumulated detail when selection of an arbitrarily chosen instance can be relied upon in every situation due to the presence of an Axiom of Choice which guarantees its retrieval from a range of possibilities. There is a received wisdom which says that this shaping of mathematics is fine and good, and that the avoidance of intricacy which this encourages leads to a minimality of expression which is characteristic of elegance in mathematics. There is also a reality, perhaps less well received, which says that all may not be quite so fine, quite so good, and that the intricacies avoided in the cause of elegance may be waiting just round the next conceptual corner to pounce on the unsuspecting. Indeed, the joys of life with the Axiom of Choice may actually limit, rather than expand, the extent to which mathematics may be developed.
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Dedicated to the memory of Japie Vermeulen.
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Mulvey, C.J. (2003). On The Geometry Of Choice. In: Rodabaugh, S.E., Klement, E.P. (eds) Topological and Algebraic Structures in Fuzzy Sets. Trends in Logic, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0231-7_13
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DOI: https://doi.org/10.1007/978-94-017-0231-7_13
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