“In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms … of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you’ll only create a larger system with its own unprovable statements. The implication is that all logical systems of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.”
Jones and Wilson, An Incomplete Education
… or as someone said somewhere else, truth trumps provability.
It’s quite a claim, isn’t it? … all logical systems of any complexity are, by definition, incomplete …
“Gödel’s Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths … It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you’ll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.”
Reminded me of Alan Watts’s declaration that ‘a tree is not made of wood, it is wood; a mountain isn’t made of stone, it is stone.’ (Alan Watts – To Speak the Truth) How can we ‘see’ our eyes when it’s our eyes that are doing the seeing?