11–20 of 41 posts
Earlier quoted context omitted.
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.
Isn'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 "…
Isn'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 "…
The first incompleteness theorem shows that, in a sense, the culprit of a logic's incompleteness is not its "simplicity" but its "complexity": if the logic is rich enough to include Peano arithmetic, then it is incomplete. Compared to mathematics in general, a complete logic system is far less powerful and cannot be used to prove nearly as many interesting things.
I 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…
A 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.
Are there any Gödel-like problems that does not involve self references?
Regarding [1], "circularity" in an uncountably infinite context seems different than "self-referential", though the latter is usually geometrically analogized as the former. I tentatively consider Yablo's Paradox to demonstrate that an ineradicable cycle of alternating truth-assignments is equivalent to an infinite (>= aleph-one) regression of alternating truth-assignments, and thus that circularity does not map coherently to self-reference in all logic systems.
[0] http://www.mit.edu/~yablo/pwsr.pdf
[1] http://ferenc.andrasek.hu/papersybprx/jcbeal_is_yablo_non_ci...
I 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 try to read Godel's Proof once a year or so. You might also be interested in Gregory Chaitin's Meta Math! The Quest For Omega, which relates the halting problem (among other things) to Godel. A word of warning though, the book is rather idiosyncratic, as you might have seen from the title.
Are there any Gödel-like problems that does not involve self references?
Yablo's Paradox [0], possibly [1]. Regarding [1], "circularity" in an uncountably infinite context seems different than "self-referential", though the latter is usually geometrically analogized as the former. I tentatively consider Yablo's Paradox to demonstrate that an ineradicable cycle of alternating truth-assignments is equivalent to an infinite (>= aleph-one) regression of alternating truth-assignments, and thus…
Imagine an infinite sequence of sentences S1, S2, S3, ...,
(S1) for all k >1, Sk is untrue
(S2) for all k >2, Sk is untrue
(S3) for all k >3, Sk is untrue
...
Are there any Gödel-like problems that does not involve self references?
Yablo's Paradox [0], possibly [1]. Regarding [1], "circularity" in an uncountably infinite context seems different than "self-referential", though the latter is usually geometrically analogized as the former. I tentatively consider Yablo's Paradox to demonstrate that an ineradicable cycle of alternating truth-assignments is equivalent to an infinite (>= aleph-one) regression of alternating truth-assignments, and thus…
This Yablo's paradox seem self referential to me. The statements are making claims about a series of statements of which they themselves are part of. I see the statements only making claims about subsequent statements... but still.
How about, rather that banning "self references", you have a more carefull phrasing requiring all structures to be fully and independently defined before you start asking questions (or making claims) about them? (No side effects from the question please). Does this avoid all Goedelish paradoxes?