Squeezing multisets into real numbers

In this paper we study the encoding (Formula presented) mapping hereditarily finite sets and hypersets - hereditarily finite sets admitting circular chains of memberships - into real numbers. The map (Formula presented) somewhat generalizes the well-known Ackermann’s encoding (Formula presented) whose co-domain is (Formula presented), to nonnegative real numbers. In this work we define and study the further natural extension of the map (Formula presented) to the so-called multisets. Such an extension is simply obtained by multiplying by k the code of each element having multiplicity equal to k. We prove that, under a rather natural injectivity assumption of (Formula presented) on the universe of multisets, the map (Formula presented) sends almost all multisets into transcendental numbers.