Ok, let’s explain what Godel was saying using the game of chess.
An axiomatic system (which is basic statements plus rules to get new statements) has two features of interest- consistency and completeness.
Consistency means you cannot get contradictory statements by following the ‘rules’.
Completeness means you complete the system by having all truths represented by the axioms and rules.
Now let’s play chess. The initial positions are the axioms and the rules are ‘logic’.
You can reach proof (many positions) by using the allowed moves, or rules, the the game.
But we can imagine proofs, or positions, that we could do that are outside the rules (logic) of the game.
Like moving your Pawn like a Rook or your King like a Bishop.
The system can represent these imagined moves but it cannot prove them using the system’s logic, or rules.
The positions are not wrong, but incomplete. The positions are correct but cannot be proven using the systems rules.
godel proved nothing
as colin leslie dean points out
godels proof is unsound as it is based on invalid principle as russell ramsey wittgenstien and text books on logic say -thus his theorem is invalid and worthless -as he has not proved it he has proved nothing
i have pointed all this out to MP and he still refused to accept it
thus saying in the face of all the evidence he has moved from philosophy to a religious committment to godel
it is the biggest fraud in mathematical history
everything dean has shown was known at the time godel did his proof but no one meantioned any of it
look
godel used the 2nd ed of PM he says
note he says he is going to use AR
but
Russell following wittgenstien took it out of the 2nd ed due to it being invalid
godel would have know that russell and wittgenstien new godel used it but said nothing
ramsey points out AR is invalid before godel did his proof godel would have know ramseys arguments ramsey would have known godel used AR but said nothing
every one knew AR was invalid
they all knew godel used it
but nooooooooooooo one said -or has said anything for 76 years untill dean
the theorem is a fraud the way godel presents it in his proof it is crap
I don’t understand why this could be fraudulent or why anyone would care so much. Godel is saying something that shouldn’t be attacked to harshly, unless it’s misunderstood. Saying that something exists outside a system that cannot be proven by the system isn’t a threat to anything. Saying the math is an incomplete language and does not adhere perfectly to reality is well accepted among mathematicians.
Let’s say Godel is wrong, what’s your point? Can you fill the gaps Godel talks about? If so, do it for me.
By the way, it is not about formal proof. It cannot be, that’s his point. You keep saying he didn’t ‘prove’ anything. You are looking for proof of the unprovable. I would say that most understand the chess analogy, they get it. You don’t need to over-analyze it and need proof of what can be understood without words, which is why I said that there are no verbal equals to his results.
So my answer is that he doesn’t have to prove anything. You either get it or you don’t.