A note on Türing's 1936

Türing’s argument that there can be no machine computing the diagonal on the enumeration of the computable sequences is not a demonstration. As well known, Türing historical article of 1936 is the result of a special endeavor focused around the factuality of a general process for algorithmic computation. As resultant formal model his famous abstract computing machine, soon called Türing machine, could be regarded to be a universal feasibility test for computing procedures. The article begins by accurately outlining the notion of computable number, that is a real number is computable only if there exists a Türing machine that writes all the sequence of its decimal extension. The abstract machine as a universal feasibility test for computing procedures is then applied up to closely examining what are considered to be the limits of computation itself, and to defining a number which is not computable. The computable numbers do not include, however, all definable numbers; and an example is given of a definable number which is not computable (230 [4]). In Section 8. Application of the diagonal process., is reached the crucial demonstration establishing some fundamental limits of computation by defining such number through a self-referring procedure. Present note shows how this procedure can not actually be regarded as a demonstration. In the following the reader is required to know Türing article together with the original notions and symbolism therein contained [4]. We recall briefly to the reader only a few of the main ones. Computing machines. If any automatic machine M prints two kinds of symbols, of which the first kind consists entirely of 0 and 1 (the others being called symbols of the second kind), then the machine will be called a computing machine. If the machine is supplied with a blank tape and set in motion, starting from the correct initial configuration, the subsequence of the symbols printed by it which are of the first kind will be called the sequence computed by the machine. Circular and circle-free machines. If a computing machine M never writes down more than a finite number of symbols of the first kind, it will be called circular. Otherwise it is said to be circle-free. A machine will be circular if it reaches a configuration from which there is no possible move, or if it goes on moving, and possibly printing symbols of the second kind, but cannot print any more symbols of the first kind. Computable sequences. A sequence is said to be computable if it can be computed by a circle-free machine.