Submitted by BernardJOrtcutt t3_1118wno in philosophy
frnzprf t1_j99qr2q wrote
Reply to comment by Masimat in /r/philosophy Open Discussion Thread | February 13, 2023 by BernardJOrtcutt
Gödel proved his incompletenes theorem. Of course that only means something when the proof is actually correct.
It's a big proof. It has something to do with the proposition "This sentence is not provable." It turns out that this sentence is neither provable nor unprovable.
Gödel also associated propositions in formal languages with Gödel-numbers. I don't know why that's necessary. You can mathematically reason about words just as well. Something with avoiding self-referenciality?
Well, Gödel actually talks about a number that represents that weird sentence.
Masimat t1_j9k1zag wrote
What would you say about Hilbert's tenth problem? Is it truly impossible?
frnzprf t1_j9l1n2o wrote
Wikipedia says it is proven to be impossible. I'm sorry, I'm too lazy to check the proof.
Mathematics is one area, where you don't have to trust the experts, but I would probably have to devote a large amount of time to understand the proof. Mathematics is not it's own field of study with professionals that do nothing else without reason. There is always a level of math that you can only understand by dedicating more time to it.
Maybe I could suggest examining where your intuition comes from, that Hilbert's tenth problem is possible to solve (i.e. there is a certain method to find solutions). Would it have weird implications if it couldn't be solved?
The naive algorithm would be to try different numbers for x, y and z until you find a solution. But you'll never know whether you will eventually find a solution or if you'll keep searching for infinity.
0² + 1 ≠ 0, (-1)² + 1 ≠ 0, 1² + 1 ≠ 0, (-2)² + 1 ≠ 0, 2² + 1 ≠ 0, ...
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