Can enumerate proofs until find a proof of statement. Assume all true statements provable. Given phi, in parallel, attempt to prove phi and not-phi. This statement is not provable. Again, provability simply refers to syntax and algorithms, which are concrete ideas that we have the tools to describe.
Input is irrelevant.
M accepts iff can prove statement. Contradiction, since proofs are sound. Couldn't prove true statement. Construction relies on TM being able to generate its own encoding. Standard hacker fun example is program that prints its own source.