Illusion and Reality

Contents I. Introduction II. Turing’s criticism of the traditional diagonal method - Inappropriate presupposition III. Turing’s proof - the essence of undecidability IV. Conclusion Appendix. Some basic concepts from Turing's 1936 paper I. Introduction Turing's 1936 paper is, as its title suggests, On Computable Numbers, with an Application to the Entscheidungsproblem [1]. The computable number is defined by Turing as the real in terms of its binary number representation computed by the computing machine (Turing Machine ), and it is used to express the integrative concept of computability (see Appendix). Thus, the computable number becomes a guiding thread through Turing's 1936 paper. The Entscheidungsproblem ( decision problem ) is derived from the 10th problem, Determination of the solvability of a diophantine equation , of the 23 mathematical problems formulated by Hilbert at the International Congress of Mathematicians in Paris in 1900 [2]. A general form