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A layperson here, I always wanted to ask this question: Godel proves that not ALL statements that are true can be proven, because here is this one example. I ge
by dandare 8y ago
A layperson here, I always wanted to ask this question: Godel proves that not ALL statements that are true can be proven, because here is this one example. I get it, it's a valid mathematical proof. But does it really say anything about other statements, except this one? Does it says anything about statements that are not self-referential? Is it possible that the incompleteness theorem really applies only to self-referential statements and says nothing about non-self-referential statements?
- icen 8y agoThe proof of the incompleteness theorem only shows you that recursively-enumerable arithmetic systems are always missing something, because the construction always holds. It doesn't make any grand statements about the structure or nature of things that are true and without proof. The second incompleteness theorem gives an example of something undecidable that's probably more interesting that the uselessly paradoxical statement in the first: it says that no system can consistently prove its own consistency.
- svantana 8y agoThat's a common technique in math proofs, to disprove a statement by providing a counterexample. But there probably are improvable statements that are not self-referential, e.g. certain statements about infinite sets. But the grander point, that Hofstadter pushes a lot in GEB, is that any sufficiently powerful symbolic system will be able to self-reference. For example, consider a programming language that is not Turing complete. You will basically need to disallow loops, because with those you can implement a virtual machine to break all your self-reference-banning rules.