111–120 of 252 posts
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https://www.smbc-comics.com/index.php?db=comics&id=2245 There are several famous mathematicians known for their avid use of amphetamines.
Not to mention the profound impact of LSD on the founders of modern computing.
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How can a statement that is unprovable be true? I always had the impression that unprovable means you could add either the statement or its negation as an axiom, and both resulting systems are as consistent as the system you started with
"This statement is false". GEB is a marvellous work that is accessible to anyone with reasonably good school grade maths. I chanced upon it by accident in the school library one day and was hooked after a few pages. Anyway the crux of the matter is that you can very carefully construct a statement about a system that can't be either proven or disproven by that system! I don't have anything like the formal knowledge t…
My take on that statement is that it is not saying anything about the world. It only refers to itself, making it a self-contained mini-universe with no relation to the real world.
So, since it's not saying anything, it's neither true nor false. Only something confusing that feels like it should have some meaning.
is here somebody who also got the impression that the book is somewhat overrated?
Every socially awkward person who obsesses about their intellect is an in-depth explainer of Goedel, Escher, Bach. It's a fun read, and clever enough, but the relation between the work of the three is pretty shallow, and if you understand only what the book contains, you haven't gotten very deep into their valuable work. The book is more an act of self-indulgence on the author's part, or at best a tribute to the bril…
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How can a statement that is unprovable be true? I always had the impression that unprovable means you could add either the statement or its negation as an axiom, and both resulting systems are as consistent as the system you started with
I'll explain it in a different way than normal. We can define a notion of complexity for any given integer as being the size of the smallest program that returns that integer. Obviously, I'm being imprecise here, but it should hopefully be clear that it is possible to get the details right and the precise nature of those details aren't going to be relevant for what follows. Now suppose we have a program that can find…
However we can write a program that does a brute force search through all proofs from ZF looking for the complexity of i. As you just showed, this program will not establish the complexity of all integers.
But how does it fail? It fails because there are programs which do not halt, that it can't prove don't halt, which if they halted COULD return i.
And therefore the complexity cannot be determined from ZF.
The prevailing philosophy of mathematics says it is well-defined. But there are alternate (entirely consistent!) philosophies under which complexity is not well-defined at all.
But then again I tried to read it when I was in high school.
So maybe I understand more in older age.
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Is there a distinction between saying a system has something that is provable and untrue and saying the system is self contradictory (and the more generalized layman interpretation that the system is wrong).
See https://en.wikipedia.org/wiki/Consistency . Under the syntactic definition of consistency, a self-contradiction simply means that a particular statement and its logical negation can both be proved in the system. That doesn’t say anything about the truth of the statement.
Why read this when GEB is its own in-depth explainer?
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It might be easiest to give a sense of what "unprovable but true" means by way of an imagined example. Goldbach's conjecture is that "every even number bigger than 2 is the sum of exactly two prime numbers", so 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, etc. For this statement to be *true* it just means that every even number there must exist two primes that add to that number. This is a statement about infinitely many integer…
what if the proofs cannot be described as finite or countable sets that does not render a straightforward application of diagonalization? What happens to Goedel’s theorem then?