 For instance, Part i of  Proposition 1.1  shows that there is no Turing machine which outputs 1 if \(T_i\) halts on \(n\) and 0 otherwise. This is thus a formulation of Turing’s well-known  unsolvability of the Halting Problem (see the entry on Turing machines).  Part ii and iii would also now be described as expressing that the sets \(\HP_T,\) \(\HP_L,\) and \(\HP_{\textit{GR}}\) are all undecidable. By taking into account the equivalence results summarized in  Section 1.6,  Proposition 1.1 thus shows that membership in these sets cannot be decided relative to any of the models in question.