We study an encoding RA that assigns a real number to each hereditarily finite set, in a broad sense. In particular, we investigate whether the map RA can be used to produce codes that approximate any positive real number to arbitrary precision, in a way that is related to continued fractions. This is an interesting question because it connects the theory of hereditarily finite sets to the theory of real numbers and continued fractions, which have important applications in number theory, analysis, and other fields.