Turing's proof
Summary of the second proof

Less than one page long, the passage from premises to conclusion is obscure.

Turing proceeds by reductio ad absurdum. He asserts the existence of a machine E, which when given the S.D (Standard Description, i.e. "program") of an arbitrary machine M, will determine whether M ever prints a given symbol (0 say). He does not assert that this M is a "computing machine".

Given the existence of machine E, Turing proceeds as follows:
punctuation:no_comma, keywords:existence, startend:end_checker