vrypan |--o--|
@vrypan.eth
IMHO, Gödel's incompleteness theorem is one of the greatest achievements of the human mind, a mathematical theorem that touches philosophy and even religion. In simple words, Gödel *proved with math*, that no matter how complete or perfect your rules of math are, there will always be things math can’t explain fully using these rules! 🧵 https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems#
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vrypan |--o--|
@vrypan.eth
Take for example the natural numbers arithmetic (yes, yes, Peano, etc, I'm trying to keep it simple). You know how addition and multiplication of natural numbers works. And you have seen in school, that we can use these basic operations to build theorems, and prove them, or prove that they are wrong. For example, - "every number divisible by 5 ends in 0 or 5": True - "there are infinitely many prime numbers": True - "if you add two prime numbers you get a prime number": False
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vrypan |--o--|
@vrypan.eth
Great. So, until Gödel, mathematicians would pick a logical sentence like the ones above and try to prove it (problem). And the assumption was that eventually, if they are smart enough, and given enough time, they could prove if it's true or false. What Gödel told them is that the sentence they are trying to prove may be true, BUT UNPROVABLE! And the worst part, they would never know if they picked one of these truths that can't be proven!
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vrypan |--o--|
@vrypan.eth
So, I could insist my whole life for example that "there are infinitely many primes p such that p+2 is also prime". As far as we know: 1. It may be true and we will be able to prove it sometime. 2. It may be false and we will be able to prove so sometime. 3. It may be true, but we may never be able to prove it is true! And for as long as we have not proven 1 or 2, 3 may be the case. But if 3 is actually the case, a) we will never know it and b) every resource we spend on trying to prove this sentence is a waste...
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vrypan |--o--|
@vrypan.eth
<end of Gödel /> <start of my thoughts> Now, if this is the case about arithmetic with simple numbers like 1,2,3,..., imagine how broken or futile may be our effort to try and explain the world with laws... Or even think that if we agree on some basic truths (axioms) about life or ethics, or politics, then we can use logic to find solutions.... <end thoughts>
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cqb
@cqb
I need to dig into the incompleteness theorem more, but does it state that a thing that is false can in principle never be proven mathematically? Or does it only apply to possibly true things?
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vrypan |--o--|
@vrypan.eth
Not claiming to be an expert and understanding the full details of the theorem is beyond my level. But here's how I think it goes: - If a statement is false, it won’t be provable. (You can't prove 1=2). - If a statement is true, it might be unprovable.
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cqb
@cqb
I'm wondering if this reinforces a popperian model of knowledge, where we can only disqualify hypotheses to get a more accurate model of the world. Another question that comes up: if we don't know if we can prove a true statement why not just negate it and try to prove the negative case instead?
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