A key part of Turing's computability paper hinges on the diagonal argument. It's really well covered in 'The Annotated Turing' by Charles Petzold. In Cantor's

A key part of Turing's computability paper hinges on the diagonal argument. It's really well covered in 'The Annotated Turing' by Charles Petzold. In Cantor's version, we prove that numbers (real numbers?) are not enumerable by creating a new number. In Turing's paper, he enumerates all valid Turing machines in a similar way, but crucially, we can't know if the new constructed machine is valid or not, so the outcome is different. It's quite possible I'm botching elements of these, but it was a beautiful read.