Abstract
We examine the notions of negative, infinite and hotter than infinite temperatures and show how these unusual concepts gain legitimacy in quantum statistical mechanics. We ask if the existence of an infinite temperature implies the existence of an actual infinity and argue that it does not. Since one can sensibly talk about hotter than infinite temperatures, we ask if one could legitimately speak of other physical quantities, such as length and duration, in analogous terms. That is, could there be longer than infinite lengths or temporal durations? We argue that the answer is surprisingly yes, and we outline the properties of a number system that could be employed to characterize such magnitudes.
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This paper grew out of a talk I gave at Northwestern University during the summer of 1977. Thanks are owed to Arthur Fine, Neal Grossman, David Malament, Brian Skyrms and Paul Teller, each of whom attended the talk and offered helpful comments as well as support. Of them I owe a special thank you to Arthur Fine with whom I have discussed many of the ideas in this paper. Finally, I am greatly indebted to Howard Stein, who brought J. H. Conway's Surreal Numbers to my attention and who suggested their relevance to my work.
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Ehrlich, P. Negative, infinite, and hotter than infinite temperatures. Synthese 50, 233–277 (1982). https://doi.org/10.1007/BF00416903
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DOI: https://doi.org/10.1007/BF00416903