You can also obtain incompleteness from the unsolvability of the halting problem, by noting that if every statement in (say) Peano arithmetic were provable, one could solve the halting problem. Encode a halting execution of a TM as an integer using Gödel numbers and write a statement that the execution halts. Either that statement or its negation would be provable, so search for proofs for each at the same time.

An additional related theorem is Rogers' recursion theorem, which is how we get programs that, when run, print their own source code (by the theorem this can be done in any Turing complete programming language.)