My incomplete theory about Gödel
I’m exploring maths, and my first step is Gödel’s incompleteness theorem as explained through the eyes of Douglas Hofstadter’s famous book “Gödel, Escher, Bach: An Eternal Golden Braid“. I still have a few chapters to go… but so far it’s a good read, and I find myself particularly looking forward to more chats between the Tortoise and Achilles. The book has given me a greater appreciation for Bach, and especially Escher… but strangely, not so much for Gödel.
First, think about the sentence “this sentence is false”. It the sentence true? If so, it must be false. But if it is false, then it must be true. But if it is true, then it must be false. But if it is false… you get the idea. It’s a paradox of language caused by a self-referential infinite loop.
Now, a similar trick can be done in number theory once you accept Gödel’s good idea that you can translate theorems into numbers. You can then develop the theorem that “this string [G] is not a theorem in formal number theory”. Is [G] telling the truth?
I think this is where the standard interpretation gets dull.
The standard interpretation is that if [G] is true, then it must not be provable in number theory. But if it’s not provable in number theory, that means number theory is incomplete because there is a true statement [G] that can’t be proven in number theory. This can’t be fixed, and hence the conclusion that number theory is forever incomplete.
Well, yeah. OK. But all this shows is that the self-referencing infinite loop paradox (“this sentence is false”) can be translated into any language, including number theory.
This certainly did put an end to David Hilbert’s second problem, where the maths gurus of the day were searching for a perfect proof for all number theory, and well done to him for that. But in my incomplete opinion, I think the focus has then shifted to the wrong issue. Instead of saying “number theory is incomplete” I think it makes more sense to say “any system of language can be broken with a self-referential infinite loop paradox”.
Consider this interpretation of [G]. If it is true, then it must be false. But if it is false, then it must be true. But if it is true, then it must be false. But if it is false… you get the idea. The standard interpretation stops this iteration after one round and concludes “the statement is true, and the system didn’t know”. But it is equally valid to say “the system if fine, the statement is neither true nor untrue… it’s just a crazy loop that is inherent to all language systems”.
The key issue is that any language can form a junk sentence with a self-referential infinite loop. It’s more of a Wittgensteinian philosophical puzzle (language trick) rather than a fundamental philosophical problem. The more interesting follow up questions then are (1) is number theory complete except for the crazy loop; and (2) what is the inherent nature of crazy loops?
And for that, perhaps Escher is the master.