Gödel and the limits of logic (2006)
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Gödel and the limits of logic (2006)
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Re: Gödel and the limits of logic (2006)
#2I highly recommend getting Godel's proof[1] and reading it. It's an amazing journey and quite understandable from the start (the first half of the book is introduction), and I've never been able to read proofs very well. It takes concentration, but once you get it (at '17 GenR') it's almost like a symphony going off. The crystalline brilliance of the "System P" decomposition is worth it just to see how math as a process could be reduced to a such simple, clear and concise set of symbol manipulations...and then the rest shows how that could be used topple itself. Incredibly philosophically insightful.
[1] https://www.amazon.ca/Undecidable-Propositions-Principia-Mat...
Re: Gödel and the limits of logic (2006)
#3Isn't Gödel's incompleteness a consequence of overly ambitious material implication properties? When you look at the truth table:
A | B | A -> B
0 | 0 | 1
0 | 1 | 1
1 | 0 | 0
1 | 1 | 1
I fail to see why all except for 1 -> 0 = 0 aren't undefined. I understand reasoning behind why this was done, but mixing unrelated predicates seems like a generally bad idea, even if it allows mechanical proofs.
Aren't there already "complete" systems like relevance logic? Now with computers we can perhaps make proofs we need "relevant" instead of being based on fundamentally incomplete though "simpler" logic?
Re: Gödel and the limits of logic (2006)
#4I like very much the program x virus example/citation.. but I would put it in the other way around:
"A program can find all virus, unless the operating system is a virus itself."
Re: Gödel and the limits of logic (2006)
#5Isn't Gödel's incompleteness a consequence of overly ambitious material implication properties? When you look at the truth table: A | B | A -> B 0 | 0 | 1 0 | 1 | 1 1 | 0 | 0 1 | 1 | 1 I fail to see why all except for 1 -> 0 = 0 aren't undefined. I understand reasoning behind why this was done, but mixing unrelated predicates seems like a generally bad idea, even if it allows mechanical proofs. Aren't there already "…
1 -> 0 = 0
Re: Gödel and the limits of logic (2006)
#6A light take on the subject can be found in Logicomix: https://en.wikipedia.org/wiki/Logicomix - a cool blend between the dry topic and comics.
Re: Gödel and the limits of logic (2006)
#7Isn't Gödel's incompleteness a consequence of overly ambitious material implication properties? When you look at the truth table: A | B | A -> B 0 | 0 | 1 0 | 1 | 1 1 | 0 | 0 1 | 1 | 1 I fail to see why all except for 1 -> 0 = 0 aren't undefined. I understand reasoning behind why this was done, but mixing unrelated predicates seems like a generally bad idea, even if it allows mechanical proofs. Aren't there already "…
1 -> 0 = 0 What does this mean? Is the 0 after the equals the antecedent? Consequent?
(1 -> 0) = 0.
In other words, (true implies false) is false.
Re: Gödel and the limits of logic (2006)
#8I highly recommend getting Godel's proof[1] and reading it. It's an amazing journey and quite understandable from the start (the first half of the book is introduction), and I've never been able to read proofs very well. It takes concentration, but once you get it (at '17 GenR') it's almost like a symphony going off. The crystalline brilliance of the "System P" decomposition is worth it just to see how math as a proc…
I can't speak for Godel's original proof since I haven't seen it before, but I certainly found Computability and Logic [1] to be pretty approachable. It's the textbook used for UC Berkeley's "Intermediate Logic" philosophy course (in other words, it's so easy even philosophy majors can understand it! :P).
From my understanding, C&L diverges from Godel's original proof technique in order to make it easier to follow, but it's much more rigorous and explicit than what you'd find in Godel, Escher, Bach or something. It's still a textbook.
[1]: https://www.amazon.com/Computability-Logic-George-S-Boolos/d...
Re: Gödel and the limits of logic (2006)
#9Isn't Gödel's incompleteness a consequence of overly ambitious material implication properties? When you look at the truth table: A | B | A -> B 0 | 0 | 1 0 | 1 | 1 1 | 0 | 0 1 | 1 | 1 I fail to see why all except for 1 -> 0 = 0 aren't undefined. I understand reasoning behind why this was done, but mixing unrelated predicates seems like a generally bad idea, even if it allows mechanical proofs. Aren't there already "…
Logic already has separate symbols for 'A implies B' and 'A proves B', where the first can be considered equivalent to '(not A) or B' for all intents and purposes. Since 'A proves B' seems like the very definition of relevance, the only part you could possibly object to is that 'A proves B' implies 'A implies B', but that's given since the alternative would lead to a contradiction where A proves B, but B is false while A is not.
Also, if I understand you correctly you're trying to 'fix' incompleteness by making A -> B undecidable, which seems like it would achieve the opposite.
Re: Gödel and the limits of logic (2006)
#10Isn't Gödel's incompleteness a consequence of overly ambitious material implication properties? When you look at the truth table: A | B | A -> B 0 | 0 | 1 0 | 1 | 1 1 | 0 | 0 1 | 1 | 1 I fail to see why all except for 1 -> 0 = 0 aren't undefined. I understand reasoning behind why this was done, but mixing unrelated predicates seems like a generally bad idea, even if it allows mechanical proofs. Aren't there already "…
Well no, Gödel's incompleteness theorem doesn't really have anything to do with material implication. It tells us that there is some formula G such that neither G nor ¬G is provable. Proof systems for relevance logic restrict valid derivations compared to classical propositional logic (relevance logic requires you to use the antecedent), so they can prove even fewer things.