Surreal dimensions

In this paper we use Conway's surreal numbers to define a refinement of the box-counting dimension of a subset of a metric space. The surreal dimension of such a subset is well-defined in many cases in which the box-counting dimension is not. Surreal dimensions refine box-counting dimensions due to the fact that the class of surreal numbers contains infinitesimal elements as well as every real number. We compute the surreal dimensions of generalized Cantor sets, and we state some open problems.