One can then use (some type of) diagonalization in combination with U to derive a contradiction. Diagonalization was launched by Cantor to point out that the set of actual numbers is “uncountable” or not denumerable. A variant of the strategy was used also by Gödel in the proof of his first incompleteness theorem.
It can also be one of the primary reasons why Turing has been retrospectivelyidentified as one of many founding fathers of computer science (see Section 5). The common Turing machine which was constructed to prove the uncomputability of certain issues, is, roughly speaking, a Turing machine that is able to compute what some other Turing machine computes. Assuming that the Turing machine notion totally captures computability (and in order that Turing’s thesis is valid), it is implied that anything which could be “computed”, can also be computed by that one common machine. Conversely, any problem that isn’t … Read More