Abstract
Starting from the set of functions which is in one-to-one correspondence with countable ordinals, and following the procedure which is analoguous to the construction of real numbers from the natural numbers, we obtain the set of transreal numbers which has the cardinality ℵ2 and interpolates the set of real numbers. Differential calculus based on this number system (in a truncated form) is presented and it is shown that a function of a transreal variable has, in general, an infinity of derivatives at a point. Representation of rotation and Lorentz group based on transreal numbers is obtained, and we find that, in addition to the smooth interpolation of the usual representation, there exist representations of both finite and infinite dimensions that are discontinuous everywhere.
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