So I've been trying to understand this theorem for the past week and there is a proof (Not rigorous) that was given to me that I just cannot understand. I'm not gonna lie I'm not so good at mathematics but, I've read it like 50 times+ and it just doesn't make sense to me.
Can anyone for the love of god help dumb this down step by step? I'm genuinely so lost... I understand godel numbers and the general structure of how it's made.
Here's the proof:
We begin with the observation that we can substitute the Godel Number of a formula into itself. Consider the statement “The formula with Godel number sub(y, y, m) cannot be proved”, where our notation sub() means to substitute, our formula has y somewhere in it and we substitute variable y anywhere y exists in the formula with the Godel Number of y that is denoted as m. This formula itself should have a Godel Number and we denote it as n.
Create new formula by substituting the number n everywhere for y in our previous formula, which now reads “The formula with Godel Number sub(n, n, 17) cannot be proved. Let’s call this formula G and has its own Godel Number. But by definition, sub(n, n, 17) is the Godel Number of the formula that results from taking the formula with Godel Number n and substituting n anywhere there’s a symbol with Godel Number 17. Because of unique prime factorization, formula G is talking about itself. So G states that it can’t be proved itself, but if proven it would mean opposite of G, which says no proof exists and thus G is undecidable. But G is true and yet undecidable That can be constructed from an axiomatic system, thus any system can be shown to be incomplete.
