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I'm no expert, but yes, I think you're wrong about that. This explanation https://www.scientificamerican.com/article/what-is-godels-theorem/ https://www.scient
by Elrac 9y ago
I'm no expert, but yes, I think you're wrong about that.
This explanation https://www.scientificamerican.com/article/what-is-godels-theorem/ https://www.scientificamerican.com/article/what-is-godels-th..., a little closer to layperson level, uses integer arithmetic as a simple example. Peano's axioms completely describe integer arithmetic - easy peasy! What Gödel says is that there are, nevertheless, statements about results in this system that cannot be proved true (or false).
The problem appears to be not describing the system but proving every possible conjecture about it.
- blackflame7000 9y agoThere are certain logical traps (paradoxes) than cannot be formalized by a computer. For example, given the statement, "This sentence is False", it is not possible to deduce a boolean value describing the sentence. Thus there is some property of the sentence that is not describable to a purely logical system. Human consciousness allow us to spot the paradox where as a polynomial search algorithm could not.
- dTal 9y agoFormalizing or describing something is not the same as "deducing a boolean value". But I can ask GNU Maxima to give me a list of all numbers that satisfy the equation x+1=x-1, and it will happily tell me that there is no such number. Maxima doesn't support self-referential propositional logic, but a solver that can identify that no truth value can support "this sentence is false" doesn't need to do anything more mystical than test both cases - no "polynomial search algorithm" required. You are suggesting that "human consciousness" is a separate, ineffable thing from a "purely logical system", but I don't see any reason a computer couldn't do what your brain is doing. You can't tell me the truth value of the sentence either.