## 2 Materials and Methods
For all \(n\in\mathbb{N}\), the set of \(n\)-tuples of natural numbers, \(\mathbb{N}^{n}\), is an atomic machine-readable language, and for all \(n,k\in\mathbb{N}\), a mapping \(G:\mathbb{N}^{n}\supseteq\to\mathbb{N}^{k},\ (m_{1},\ldots,m_{n})\mapsto G(m_{1}, \ldots,m_{n})\) is called atomically computable if there exist functions \(g_{1},\ldots,g_{k}\in\mathcal{U}(n)\) such that \(\text{dom}(G)\) is a (possibly improper) subset of \(\text{dom}(g_{1})\cap\ldots\cap\text{dom}(g_{k})\), and we have  
\[G(m_{1},\ldots,m_{n})=\big{(}g_{1}(m_{1},\ldots,m_{n}),\ldots,g_{k}(m_{1}, \ldots,m_{n})\big{)}\]for all \((m_{1},\ldots,m_{n})\in\text{dom}(G)\). A (_non_-atomic) machine-readable language for the (abstract) set \(\mathcal{A}\) is of the form  
\[\mathfrak{A}:=\Big{\{}\Big{(}a,\Lambda_{\mathfrak{A}},[\cdot]_{\mathfrak{A}} \Big{)}:a\in\mathcal{A}\Big{\}},\]  
where \(\Lambda_{\mathfrak{A}}\) is a machine-readable language and \([\cdot]_{\mathfrak{A}}:\Lambda_{\mathfrak{A}}\supseteq\rightarrow\mathcal{A}\) is a partial surjective mapping. Further, \(\lambda\in\text{dom}([\cdot]_{\mathfrak{A}})\) is called a machine-readable description of \([\lambda]_{\mathfrak{A}}\in\mathcal{A}\). Again, \"\([\cdot]_{\mathfrak{A}}\)\" hints towards the usual notation for quotient sets: for \(\lambda_{1},\lambda_{2}\in\text{dom}([\cdot]_{\mathfrak{A}})\), we have \(\lambda_{1}\equiv\lambda_{2}\) if and only if \([\lambda_{1}]_{\mathfrak{A}}=[\lambda_{2}]_{\mathfrak{A}}\), i.e., \(\lambda_{1}\) and \(\lambda_{2}\) are two machine-readable descriptions of the same abstract object. However, since \([\cdot]_{\mathfrak{A}}\) is generally partial, so is the induced equivalence relation. Finally, mapping \(G_{1}:\mathfrak{A}\supseteq\rightarrow\mathfrak{B}\), where \(\mathfrak{A}\) and \(\mathfrak{B}\) are machine-readable languages, is called (_non_-atomically) computable if there exists a computable mapping \(G_{2}:\Lambda_{\mathfrak{A}}\supseteq\rightarrow\Lambda_{\mathfrak{B}}\) such that, for all \(\lambda\in\text{dom}([\cdot]_{\mathfrak{A}})\) with \(([\lambda]_{\mathfrak{A}},\Lambda_{\mathfrak{A}},[\cdot]_{\mathfrak{A}})\in \text{dom}(G_{1})\), we have  
\[[\cdot]_{\mathfrak{M}}^{n}:\mathbb{N}\to\mathcal{U}(n),M\mapsto[M]_{\mathfrak{ M}}^{n}:=U_{M}^{n},\]0  
Unless defined otherwise, a sequence \((\mathfrak{a}_{n})_{n\in\mathbb{N}}\) of elements of \(\mathfrak{A}\) is called computable if there exists a (total) computable mapping \((n\mapsto\mathfrak{a}_{n})\). Observe that if \(\mathfrak{A}\) and \(\mathfrak{B}\) are machine-readable languages for arbitrary abstract sets \(\mathcal{A}\) and \(\mathcal{B}\), respectively, \(G_{1}\) naturally induces a mapping \(G_{1}:\mathcal{A}\supseteq\rightarrow\mathcal{B}\) according to  
\[[\cdot]_{\mathfrak{M}}^{n}:\mathbb{N}\to\mathcal{U}(n),M\mapsto[M]_{\mathfrak{ M}}^{n}:=U_{M}^{n},\]1  
and vice versa. If, according to the specific context, there is no danger of ambiguity, we will not distinguish between \(G_{1}:\mathfrak{A}\supseteq\rightarrow\mathfrak{B}\) and \(G_{1}:\mathcal{A}\supseteq\rightarrow\mathcal{B}\). In essence, a machine-readable language is a formal specification of how to represent abstract information on digital hardware, such that we can (in principle) trace this specification down to the level of tuples of natural numbers and fundamental operations thereon. Upon fixing a suitable \(\mu\)-recursive pairing function (Chapter 1.4, p. 12 in [16]), i.e., a bijective mapping  
\[[\cdot]_{\mathfrak{M}}^{n}:\mathbb{N}\to\mathcal{U}(n),M\mapsto[M]_{\mathfrak{ M}}^{n}:=U_{M}^{n},\]2  
with \(\langle\cdot\rangle_{1},\langle\cdot\rangle_{2}\in\mathcal{U}(1)\), every machine-readable language \(\mathfrak{A}\) exhibits a canonical numbering, i.e., computable surjective mapping \(\varphi_{\mathfrak{A}}:\mathbb{N}\supseteq\rightarrow\mathfrak{A}\), defined in an inductive manner:  
* If \(\mathfrak{A}\) is an atomic machine-readable language, i.e., we have \(\mathfrak{A}=\mathbb{N}^{n}\) for some number \(n\in\mathbb{N}\), we define \[\varphi_{\mathfrak{A}}(m):=