Here by 'provable' and 'unprovable' we don't mean absolutely unprovable (i.e. not provable from any set of axioms) since there are no absolutely unprovable formulas. Rather we mean unprovable in whatever theory is in question (let's say it's PA for simplicity of exposition). — TonesInDeepFreeze
correct me if I'm wrong, given an axiomatic "theory" A, and Godel sentence G = the theorem T is true and unprovable in axiomatic theory A". — TheMadFool
G is claiming that T is true in A — TheMadFool
T isn't true in some other "theory" like you seem to suggesting when you say "Rather what we mean unprovable in whatever theory is in question" but in A. — TheMadFool
G is unprovable in A.
And G does not say
G is true and unprovable in A — TonesInDeepFreeze
G is unprovable in A.
And G does not say
G is true and unprovable in A — TonesInDeepFreeze
In my view Gödel's incompleteness theorems, as the other incompleteness results, aren't roadblocks.It has the respect of most in the math community, but most of those think they will never come up against that roadblock. — jgill
It has the respect of most in the math community, but most of those think they will never come up against that roadblock. — jgill
In my view Gödel's incompleteness theorems, as the other incompleteness results, aren't roadblocks. — ssu
"The first incompleteness theorem shows that the Gödel sentence G_F of an appropriate formal theory F is unprovable in F. Because, when interpreted as a statement about arithmetic, this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true. For this reason, the sentence G_F is often said to be "true but unprovable." However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence GF may only be arrived at via a meta-analysis from outside the system." — TonesInDeepFreeze
statement GF in F is often called “the Gödel sentence” of F — SEP
Therefore (1), GF cannot be false, and must be true. For this reason, the Gödel sentence is often called “true but unprovable (2)” — SEP
Godel sentence = G = There's a mathematical theorem, call it T, in a given axiomatic system A, such that T is unprovable/undecidable (which word is apt?) in A. — TheMadFool
Kurt Godel's tour de force was proving G, the Godel sentence, is true. Am I right? — TheMadFool
Therefore (1), GF cannot be false, and must be true. For this reason, the Gödel sentence is often called “true but unprovable (2)”
— SEP
The word "therefore" (1) suggests an argument i.e. there's a proof for the Godel sentence GF. However, the next line asserts that GF is ...often called "true but unprovable (2)". — TheMadFool
the unprovable status of GF is not what matters — TheMadFool
GF should be asserting a mathematical theorem, call it T, and asserting that T is unprovable and not that GF itself can't be proved. — TheMadFool
No.Kurt Godel's tour de force was proving G, the Godel sentence, is true. Am I right? — TheMadFool
you really would need to read a book in mathematical logic — TonesInDeepFreeze
It works, roughly, like this: Godel created/discovered a method by which every proposition and every sequence of propositions in T can be assigned a unique number. His sentence (a number - a pretty big number) then (when translated appropriately), becomes (roughly), "The proposition with the Godel number G is not provable (in T)." And you might ask, so what? Well, the number of this proposition is just G itself! (How did he do that? Read the paper, or research Godel numbering. He did it his way, and subsequently other people found different ways.) — tim wood
To grasp how exactly it all works and makes perfect and rigorous sense, you really would need to read a book in mathematical logic — TonesInDeepFreeze
I'll call that statement C* (it is a part of the incompleteness theorem C). Both C and C* are not stated in the language of F, but rather in a meta-language for F. And neither C nor C* are the Godel-sentence. In other words, the statement of the incompleteness theorem is different from the Godel-sentence that is used to prove the incompleteness theorem. — TonesInDeepFreeze
But the fact remains that math people not in those areas are usually not very concerned, even if they are stumped in proving something. However, I haven't been around mathematicians for a long time and I could be mistaken. — jgill
Yes, it is. Now. Either you satisfy yourself with the level of understanding that English sentence, "This sentence is false," provides, or you do some reading. .[The proposition with Gödel number G is not provable] is definitely not a mathematical theorem. — TheMadFool
Yes, it is. Now. Either you satisfy yourself with the level of understanding that English sentence, "This sentence is false," provides, or you do some reading. . — tim wood
[The proposition with Gödel number G is not provable] is definitely not a mathematical theorem. What gives? — TheMadFool
This Numberphile video might help:
https://youtu.be/O4ndIDcDSGc
(Around 5:32 he starts to talk about Gödel numbering) — Amalac
Yes, he himself says he is simplifying it, but I does give one a rough idea about it, which I thought was a good start for TheMadFool. — Amalac
For instance, at one phase, it asserts that the provability of a theorem hinges on its Godel number being divisible, by the Godel numbers of a formalized system's constituent axioms. — Aryamoy Mitra
the key proposition in Gödel's proof is K = The proposition with Gödel number G is not provable (in T) and the "coincidence" is that K is the proposition with Gödel number G. In other words, K is not provable. — TheMadFool
has to be a mathematical theorem, no? — TheMadFool
In the same way that words have to be spelled correctly. Except that as a practical matter, words do not really have to be spelled correctly - colse cna wrok. But not in math. To say that thirteen is almost divisible by four is to say that thirteen is not divisible by four, and sometimes the does it or doesn't it is what matters. — tim wood
Btw, I see above you're reading an online paper of about 150+ pages, Godel's theorem itself is about 34 pages, and very readable. Even a non-math person like me, with a little effort and work, can get most of it. Of course there are subtleties, like the depth of water under the ice you're skating on. But if the ice is good and skating is what you're doing, then why break through it? — tim wood
No such thing as a short translation. But a translation exists here, Godel's pp. 4-39, but other treasures in the book, e,g., pp. 305-337, by Emil Post, although this one starts easy and gets hard.:I've searched comprehensively for short translations of Gödel's 1931 paper, to no success. — Aryamoy Mitra
If the notion of Gödel divisibility was of an exact accuracy, though, one might witness a multitude of surprising repercussions emerge. Wouldn't it then be plausible, to Gödel code every unsolved hypothesis - and reverse-trace an axiomatic proof? — Aryamoy Mitra
short translations — Aryamoy Mitra
No such thing as a short translation. But a translation exists here, Godel's pp. 4-39, but other treasures in the book, e,g., pp. 305-337, by Emil Post, although this one starts easy and gets hard.:
https://www.amazon.com/Undecidable-Propositions-Unsolvable-Computable-Mathematics/dp/0486432289/ref=sr_1_1?dchild=1&keywords=The+Undecidable%2C+davis&qid=1616602363&s=books&sr=1-1
And cheaper here:
https://www.abebooks.com/servlet/SearchResults?cm_sp=SearchF-_-topnav-_-Results&ds=20&kn=the%20undecidable%2C%20davis&sts=t — tim wood
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