Eli Mellen08/11/2023, 6:57 PM
Imagine a novel, which is imbedded in a longer novel, which itself is imbedded in an even longer novel, and so on ad infinitum. Within each novel, the characters can debate the literary merits of any of the sub-novels. But, by analogy with classes of machines that can’t analyze themselves, the characters can never critique the novel that they themselves are in. (This, I think, jibes with our ordinary experience of novels.) To fully understand some reality, we need to go outside of that reality. This is the essence of Kleene’s hierarchy: that to solve the Halting Problem for some class of machines, we need a yet more powerful class of machines.
Andrew F08/12/2023, 11:34 PM