Does this actually mirror the incompleteness theorem logic or does it just create an infinite recursive loop?
They seem distinct.
They seem distinct.
The argument given here is a modern translation of Turing's work in his original paper introducing Turing Machines and the Halting problem, in which he devotes an entire section to the connection between his proof and that of Godel.
It says there are something non-mechanical about intelligence.
Apparently people think it is just a paradox or some other boring idea.
See "Emperor's New Mind" from Roger Penrose.
I don't think Gödel drew that conclusion; that's Roger Penrose's reasoning. He uses Gödel's Theorem as an example of correct reasoning that can't be algorithmic (i.e. Gödel's mind cannot be replaced by an algorithm). It's not a proof, and it's controversial. I find it convincing; but then I'm biased - I'm predisposed to the view that the mind is not algorithmic.
But I find it very disheartening when such an interesting find is presented as simply "a paradox" akin to the phrase "this is a lie". That would be the most uninteresting thing ever.
Many articles on the internet thus describe Gödel's theorem, and also that famous book from Douglas Hofstadter.
Mayhaps the issue is approaching the problem at the same dimensional level that the problem originated from. After all, humans can come at the problem from a higher cognitive level and, given enough time, resolve that an infinite loop is taking place, so the problem is not fundamentally unsolvable.
Einstein said, "No Problem Can Be Solved From The Same Level Of Consciousness That Created It", so it is feasible to assume that, say, a quantum computer running every possible iteration of an algorithm could easily identify which iterations of the algorithm's parameters would induce infinite loops by simple attrition.
Hofstaedter discusses this at length in his book Gödel, Escher, Bach if you are interested in this topic.