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I found this nice presentation about Godel's theorems. https://youtu.be/HeQX2HjkcNo?t=895 https://youtu.be/HeQX2HjkcNo?t=895 "There is no proof for the stateme

I found this nice presentation about Godel's theorems. https://youtu.be/HeQX2HjkcNo?t=895 https://youtu.be/HeQX2HjkcNo?t=895 "There is no proof for the statement with Godel-number g". What I'm struggling with is trying to understand if self-reference can be formally represented by some symbol. Can it? Godel's proof seems to rely on a trick that allows a formal system to talk about itself but is that possible? There would need to be a symbol for that and thus a symbol that refers to itself? A picture of a pipe is not a pipe. Similarly a representation of a proof is not a proof, is it?. Not saying I believe I'm right and Godel wrong, but it just feel strange to me that a formal system could ever "prove" anything about itself. What does it mean that something is "true"? That it can be proven?

zuminator 5y ago
Not every picture of a pipe is a pipe but it's at least conceivable that an actual working pipe can be constructed out of something which happens to be a picture of a pipe. Similarly Godel isn't proving that all statements in a system are paradoxically self-referential. Only that given any system of sufficient complexity, one can conceivably be constructed. In any event, all of our mathematical proofs are predicated upon the idea that the representation of a mathematical object can be treated in terms of proof as equivalent to the mathematical object. Is "2" the number 2 or is it just a glyph that non-representationally depicts the successor of the first counting number? And if you believe the latter, then how do you go about proving anything about the actual successor of the first counting number without resorting to glyphs?
- galaxyLogic 5y ago
  > given any system of sufficient complexity, That's my question. It seems that adding "self reference" to the system makes it more complicated than it would be otherwise. So it's not just a system with enough complexity to let you do arithmetic. You need a system with enough complexity to also do self-reference. And I would say and think that self-reference is complicated. Do we need self-reference in any other areas of mathematics, than Godel's and related proofs?

zuminator 5y ago

Not every picture of a pipe is a pipe but it's at least conceivable that an actual working pipe can be constructed out of something which happens to be a picture of a pipe. Similarly Godel isn't proving that all statements in a system are paradoxically self-referential. Only that given any system of sufficient complexity, one can conceivably be constructed. In any event, all of our mathematical proofs are predicated upon the idea that the representation of a mathematical object can be treated in terms of proof as equivalent to the mathematical object. Is "2" the number 2 or is it just a glyph that non-representationally depicts the successor of the first counting number? And if you believe the latter, then how do you go about proving anything about the actual successor of the first counting number without resorting to glyphs?

galaxyLogic 5y ago
> given any system of sufficient complexity, That's my question. It seems that adding "self reference" to the system makes it more complicated than it would be otherwise. So it's not just a system with enough complexity to let you do arithmetic. You need a system with enough complexity to also do self-reference. And I would say and think that self-reference is complicated. Do we need self-reference in any other areas of mathematics, than Godel's and related proofs?