Listed as 402 pages. Godel's paper is 34 pages. Interestingly, I find only one place where (in my translation) Godel uses "true" or variants: section 3, "The following proposition is true: Theorem VII: Every recursive relation is arithmetical." In the rest it's "provable" or "decidable," and variants.An Introduction to Godel's Theorems" by Peter Smith — sime
Listed as 402 pages. Godel's paper is 34 pages. Interestingly, I find only one place where (in my translation) Godel uses "true" or variants: section 3, "The following proposition is true: Theorem VII: Every recursive relation is arithmetical." In the rest it's "provable" or "decidable," and variants. — tim wood
The usual locution I find is that G is undecidable, but because G says it's undecidable, it's true; this a so-called metamathematical proof being outside the system in which G is created. — tim wood
Why is this a reasonable idea? I can assign numbers to anything. "Leaves" is 3. "Are" is 4. "Green" is 5. "Leaves are green" is 3 4 5. Now apply those numbers to the first primes, and multiply:As to why, he wishes to "encode" propositions and proofs as numbers so that he can say, e.g., x B y, meaning that there is a relationship between the variables x and y — tim wood
Many mathematicians and logicians cannot themselves be bothered to master Godel's incompleteness proof, for there isn't any payoff for doing so, and will probably content themselves with a technical understanding of the weaker version I mentioned above that is straightforward to prove and remember, and loses practically nothing — sime
That's easy. Our language allows for sentences to refer to themselves, as Hofstadter demonstrated:How do you get a sentence to be inside of itself? — tim wood
This is what's tripping me up. Take the number 144. It doesn't refer to itself. Although it is the square of 12, it doesn't mean the square of 12, or refer to squares in general. 144 is also the sum of the the primes 47 and 97, but it doesn't mean the sum of those two primes, or refer to the idea of even numbers > 2 being the sum of two primes. And it doesn't refer anything else.Or, what Godel did, was to get a number to refer to itself while at the same time referring to propositions and arguments. — tim wood
That's easy. Our language allows for sentences to refer to themselves, as Hofstadter demonstrated:
'“Is white” is white.' — Patterner
The answer is just 30-odd pages away, actually in the first five pages, sec. 1, in sketch form, with some effort on your part. And not some three- or four-hundred page book.What am I not understanding? How did Godel make numbers self-referential? — Patterner
Ok, I guess that's not referring to itself. Maybe, "This sentence is referring to itself." Or, in a more informative way, "This sentence has five words." Incorrectly, "This sentence had eighty three words."Not quite. What is your original sentence? — tim wood
I assume you mean his original paper? Any particular translation?The answer is just 30-odd pages away, actually in the first five pages, sec. 1, in sketch form, with some effort on your part. And not some three- or four-hundred page book. — tim wood
Either G is provable or not provable — TheMadFool
1. G is provable. So G is unprovable
2. G is not provable
So, there is G in the theory T
Have I got it right? — TheMadFool
For me the problem starts with 'This sentence is not provable'. This is meaningless. It does not state what is not provable. — FrancisRay
G is a number constructed following explicit rules. — tim wood
I need to find a professor of mathematics to sit down with me and help me understand it. — Patterner
"The statement on the list at location G is not provable." — tim wood
It would help if I could find a statement that is true but provably undecidable, but I've never seen one. Do you have an example? — FrancisRay
[italics in original]Godel's completeness conjecture — sime
I guess there's no hope for me in this. — Patterner
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