## 2 Materials and Methods
In general, an explicit definition of a mapping is of the form \(G:\mathcal{A}\supseteq\mathcal{B}\), \(a\mapsto G(a):=``\text{EXPR}(a)\"\), where \(\mathcal{A}\) and \(\mathcal{B}\) are arbitrary sets, and \"\(\text{EXPR}(a)\"\) is the term defining \(G\), such as, for example, \"\(a+a\)\", \"\(a^{2}\)\", \"\(\mathds{1}(a)\)\", and so on. If the context determines \(\mathcal{A}\) and \(\mathcal{B}\) without ambiguity, but does not require providing an explicit definition, we simply write \((a\mapsto\text{``EXPR}(a)\text{''})\) to denote the respective mapping. The formalization of how to represent information on Turing machines builds upon the existence of universal \(\mu\)-recursive functions \(U:\mathbb{N}\times\mathbb{N}\supseteq\to\mathbb{N}\), an arbitrary one of which we fix for the remainder of this article. Then, for every \(\mu\)-recursive function \(g:\mathbb{N}^{n}\supseteq\to\mathbb{N},\ n\in\mathbb{N}\), the following holds true: there exists a \"program\" \(M\in\mathbb{N}\) such that the function \(U_{M}^{n}:\mathbb{N}^{n}\supseteq\to\mathbb{N},\ (m_{1},\ldots,m_{n})\mapsto U_{M}^{n}(m_{1}, \ldots,m_{n})\), defined through  
\[U_{M}^{n}(m_{1},\ldots,m_{n}):=U(\cdots U(U(M,m_{1}),m_{2})\cdots,m_{n})\]  
satisfies \(g=U_{M}^{n}\). Accordingly, for all \(n\in\mathbb{N}\), the universal \(\mu\)-recursive function \(U\) provides an equivalence relation on \(\mathbb{N}\): for \(M,K\in\mathbb{N}\), we have \(M\equiv K\) if \(U_{M}^{n}=U_{K}^{n}\). Evidently, the equivalence-relation's quotient set \(\{\{K\in\mathbb{N}:K\equiv M\}:M\in\mathbb{N}\}\) is in one-to-one correspondence with the the set \(\mathcal{U}(n)\). We denote  
\[[\cdot]_{\mathfrak{M}}^{n}:\mathbb{N}\to\mathcal{U}(n),M\mapsto[M]_{\mathfrak{ M}}^{n}:=U_{M}^{n},\]  
which hints towards the usual notation for quotient sets in the context of equivalence relations: we have \([M]_{\mathfrak{M}}^{n}=[K]_{\mathfrak{M}}^{n}\) if and only if \(M\equiv K\). We call the set  
\[\mathfrak{U}(n):=\Big{\{}\left(g,\mathbb{N},[\cdot]_{\mathfrak{M}}^{n}\right) :g\in\mathcal{U}(n)\Big{\}}\]  
a machine-readable language for the set \(\mathcal{U}(n)\), and \(M\in\mathbb{N}\) is a machine-readable description of \(U_{M}^{n}\in\mathcal{U}(n)\). Furthermore, for \(n,m\in\mathbb{N}\), a mapping \(G:\mathfrak{U}(m)\supseteq\to\mathfrak{U}(n)\) is called computable if there exists a \(\mu\)-recursive function \(g:\mathbb{N}\supseteq\to\mathbb{N}\) such that, for all \(M\in\mathbb{N}\) with \(U_{M}^{m}\in\text{dom}(G)\), we have  
\[G\Big{(}h,\mathbb{N},[\cdot]_{\mathfrak{M}}^{m}\Big{)}=\Big{(}U_{g(M)}^{n}, \mathbb{N},[\cdot]_{\mathfrak{M}}^{n}\Big{)}.\]  
For \(m_{1},\ldots,m_{k}\in\mathbb{N}\), we can extend the principle to analogously mappings of the form \(G:\mathfrak{U}(m_{1})\times\ldots\times\mathfrak{U}(m_{k})\supseteq\to \mathfrak{U}(n)\). Observe that arithmetic operations on \(\mu\)-recursive functions, such as  
\[(g_{1},g_{2}) \mapsto\big{(}n\mapsto g_{1}(n)+g_{2}(n)\big{)},\] \[(g_{1},g_{2}) \mapsto\big{(}n\mapsto g_{1}(n)g_{2}(n)\big{)},\] \[(g_{1},g_{2}) \mapsto\big{(}n\mapsto\max\{g_{1}(n),g_{2}(n)\}\big{)},\] \[(g_{1},g_{2}) \mapsto\big{(}n\mapsto\min\{g_{1}(n),g_{2}(n)\}\big{)},\]  
and so on, as well as composition, primitive recursion, and unbounded search (see above), when seen as operations on \(\mu\)-recursive functions, provide computable mappings in the sense of the definition above. Throughout the remainder of the article, we will make implicit use of the computability of mappings of the form \(G:\mathfrak{U}(m_{1})\times\ldots\times\mathfrak{U}(m_{k})\supseteq\to \mathfrak{U}(n)\) on many occasions. For details, we refer to the SMN-Theorem, c.f. Theorem 3.5, p. 16 in [14]. Following the principle of \(\mathfrak{U}(n)\), \(n\in\mathbb{N}\), we can now define general machine-readable languages and general computable mappings through an inductive scheme.