Negative,infinite, and hotter than infinite temperatures

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.