Damn fine write up. Favorited this one, good job!
What Gödel Discovered
11–20 of 271 posts
Re: What Gödel Discovered
#12Thanks for great write-up, fun to read! I only notices one tiny typo: "when apples are a fruit, then bananas or applies implies bananas or fruits" - "applies" should be "apples" I guess.
Re: What Gödel Discovered
#13> This proof showed that “1 + 1”, does indeed equal “2”. It took 2 volumes to get here. I know this seems logical to mathematicians, but it feels to me like having to take 2 volumes to prove something than any child knows intuitively is... I don't know what word I am looking for... obsessive?
The idea is that 1+1 proof is made without the mathematical apparatus(High-level language) available after 1+1, i.e. its like a huge Assembler bootloader made from verified primitives(logical primitives) to load an operating system with High-level API(Mathemathics).
Re: What Gödel Discovered
#14This is remarkably well done. I'm reading Hofstadter's Godel, Escher, Bach and found this at the perfect time. Thanks for the write up!
Man, I just searched about that book, and it is like there is a world I don't know anything about. Would you mind sharing the names of your favorite books?
Re: What Gödel Discovered
#15Thanks for great write-up, fun to read! I only notices one tiny typo: "when apples are a fruit, then bananas or applies implies bananas or fruits" - "applies" should be "apples" I guess.
And "pairs" should be "pears"!
Re: What Gödel Discovered
#16Something I've always been curious about: does Gödel's theorem imply an infinity of inconsistent statements, or is the "this statement cannot be proven..." statement the only one? If it's the latter, then of what practical significance is the singularity? If a system is incomplete only in that regard, couldn't one redefine incompleteness to exclude it, and render the system complete for all practical purposes?
Re: What Gödel Discovered
#17> This proof showed that “1 + 1”, does indeed equal “2”. It took 2 volumes to get here. I know this seems logical to mathematicians, but it feels to me like having to take 2 volumes to prove something than any child knows intuitively is... I don't know what word I am looking for... obsessive?
we mostly live our lives as if P!=NP, but that does not detract from the importance of proving it.
Re: What Gödel Discovered
#18> This proof showed that “1 + 1”, does indeed equal “2”. It took 2 volumes to get here. I know this seems logical to mathematicians, but it feels to me like having to take 2 volumes to prove something than any child knows intuitively is... I don't know what word I am looking for... obsessive?
The point was to prove that math wasn't an arbitrary human construct, that it was rooted in the fundamental nature of reality.
Re: What Gödel Discovered
#19Something I've always been curious about: does Gödel's theorem imply an infinity of inconsistent statements, or is the "this statement cannot be proven..." statement the only one? If it's the latter, then of what practical significance is the singularity? If a system is incomplete only in that regard, couldn't one redefine incompleteness to exclude it, and render the system complete for all practical purposes?
From Wikipedia [0]:
> The proof constructs a particular Gödel sentence for the system F, but there are infinitely many statements in the language of the system that share the same properties, such as the conjunction of the Gödel sentence and any logically valid sentence.
[0] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...
Re: What Gödel Discovered
#20Something I've always been curious about: does Gödel's theorem imply an infinity of inconsistent statements, or is the "this statement cannot be proven..." statement the only one? If it's the latter, then of what practical significance is the singularity? If a system is incomplete only in that regard, couldn't one redefine incompleteness to exclude it, and render the system complete for all practical purposes?
if P is (this statement cannot be proved), then you can't prove A && P for any provable A, either. There's an infinite number of statements that can be proven (1=1, 2=2, 3=3... for starters) so there's an infinite number of statements that can't be proven.
There are less trivial classes of statements that can't be proven. For instance, there's an infinite number of statements like "bit N of Chaitin's constant is 1" that cannot be proven (only a finite number of those statements are provable).
https://en.wikipedia.org/wiki/Chaitin%27s_constant#Incomplet...