71–80 of 252 posts
Earlier quoted context omitted.
commitment to the material in that depth is not the product of drugs
https://www.smbc-comics.com/index.php?db=comics&id=2245 There are several famous mathematicians known for their avid use of amphetamines.
> Gödel's Incompleteness Theorem: any sufficiently rich formal system, together with an interpretation, has strings which are true but unprovable. This is only half of it! Gödel's Incompleteness Theorem states that any sufficiently rich formal system, together with an interpretation, either has strings which are true but unprovable or has strings which are provable but untrue. Either is possible! In practice people p…
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
> Gödel's Incompleteness Theorem: any sufficiently rich formal system, together with an interpretation, has strings which are true but unprovable. This is only half of it! Gödel's Incompleteness Theorem states that any sufficiently rich formal system, together with an interpretation, either has strings which are true but unprovable or has strings which are provable but untrue. Either is possible! In practice people p…
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).
Is there a good book that is similar in spirit, but doesn't require the maturity of GEB? I know an 8th grader that would be a great target, but I don't know if they have the mathematical/logical maturity to get through it.
Earlier quoted context omitted.
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…
I've read GEB over many years rather in the way someone would read the Bible. I pick it up from time to time and enjoy chewing on one or two chapters of material. But I've yet to figure out if the book actually has a specific thesis. I know it's all about the power of interpretation and the way in which interpreting a formal system as self-referencing has the effect of completely blowing up the intended design of tha…
The central thesis of GEB is this: what is a self? From the preface of the 20th anniversary edition: "GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter. What is a self, and how can a self come out of stuff that is as selfless as a stone or a puddle?"
Earlier quoted context omitted.
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…
Close but not quite as that's an inconsistent statement.
"This statement is unprovable." is the approach Godel takes and eliminates the inconsistency. Either that statement is true, in which case it's unprovable, or it's false in which case there exists a proof of a false statement.
> Gödel's Incompleteness Theorem: any sufficiently rich formal system, together with an interpretation, has strings which are true but unprovable. This is only half of it! Gödel's Incompleteness Theorem states that any sufficiently rich formal system, together with an interpretation, either has strings which are true but unprovable or has strings which are provable but untrue. Either is possible! In practice people p…
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
Kruskal's tree theorem is a more interesting example, you should look into that. You can prove it undecidable, so you can add either it or it's negation like you say, in Peano's arithmetic, but you can prove it ZFC or I think even less expressive axiomatic systems for set theory.
Am I the only one who did not find this book that interesting? I studied CS so it just felt like reading my class textbooks again, except with random trippy stories in between that try to shoehorn theory into a poor metaphor. The fundamentals of CS (strings, automata, graphs) are elementary building blocks. This is by design. You can apply them to almost anything. Almost everything "is a graph", or "recursion" if you…
Its as much a weird work of nerd art than a manual, like _whys guide to ruby_.
I've read GEB over many years rather in the way someone would read the Bible. I pick it up from time to time and enjoy chewing on one or two chapters of material. But I've yet to figure out if the book actually has a specific thesis. I know it's all about the power of interpretation and the way in which interpreting a formal system as self-referencing has the effect of completely blowing up the intended design of tha…
Godel, quines, the phonograph ship of theseus stuff, Bach harmonies, all the self-aware dialogues. He's looking for the vital spark, the thing that makes us greater than the sum of parts. But he can't capture the essence (who can?), and settles for running around the outskirts.