formula_33 ("the result of applying the instructions coded by" formula_34 "to the input" formula_35 ), and is universal in the sense that any calculable partial function formula_36 is given by formula_37 for all formula_38 where formula_39 codes the instructions for formula_2. Using the above notation formula_41, and the diagonal function arises quite naturally as formula_42. Ultimately, these ideas are connected to Church's thesis that says the mathematical notion of computable partial functions is the "correct" formalization of an effectively calculable partial function, which can neither be proved or disproved. Church used Lambda calculus, Turing an idealized computer, and later Emil Post in his approach, all of which are equivalent.