Squeezing multisets into real numbers

In this paper we study the encoding $\mathbb{R}_A(x) = \sum_{y\in x} 2^{-\mathbb{R}_A(y)}$, mapping hereditarily finite sets and hypersets - hereditarily finite sets admitting circular chains of memberships - into real numbers. The map $\mathbb{R}_A$ somewhat generalizes the well-known Ackermann's encoding $\mathbb{N}_A(x) = \sum_{y\in x} 2^{\mathbb{N}_A(y)}$, whose co-domain is $\mathbb{N}$, to nonnegative real numbers. In this work we define and study the further natural extension of the map $\mathbb{R}_A$ 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 $\mathbb{R}_A$ on the universe of multisets, the map $\mathbb{R}_A$ sends almost all multisets into transcendental numbers.