Decision problems are represented as sets consisting of objects which can serve as the inputs for a machine \(M \in \mathfrak{M}\). For instance, if \(\mathfrak{T}\) is used as the reference model then it is assumed that all problems \(X\) are represented as sets of finite binary strings – i.e. \(X \subseteq \{0,1\}^*\). This is accomplished by defining a mapping \(\ulcorner \cdot \urcorner:X \rightarrow \{0,1\}^*\) whose definition will depend on the type of objects which comprise \(X\). For instance, if \(X \subseteq \mathbb{N}\), then \(\ulcorner n \urcorner\) will typically be the binary numeral representing \(n\). And if \(X\) is a subset of \(\text{Form}_{\mathcal{L}}\) – i.e. the set of formulas over a formal language \(\mathcal{L}\) such as that of propositional logic – then \(\ulcorner \phi \urcorner\) will typically be a (binary) Gödel numeral for \(\phi\). Based on these conventions, problems \(X\) will henceforth be identified with sets of strings \(\{\ulcorner x \urcorner : x \in X\} \subseteq \{0,1\}^*\) (which are often referred to as languages) corresponding to their images under such an encoding.