Anonymous ID: f8c3a9 Aug. 1, 2022, 8:36 a.m. No.16944877   🗄️.is đź”—kun

>>16944478

think Y.

think 17.

think logically.

The formula with Gödel number sub(y, y, 17) cannot be proved.” Recalling the notation we just learned, the formula with Gödel number sub(y, y, 17) is the one obtained by taking the formula with Gödel number y (some unknown variable) and substituting this variable y anywhere there’s a symbol whose Gödel number is 17 (that is, anywhere there’s a y).

“The formula with Gödel number sub(y, y, 17) cannot be proved” — is sure to translate into a formula with a unique Gödel number. Let’s call it n.

 

Now, one last round of substitution: Gödel creates a new formula by substituting the number n anywhere there’s a y in the previous formula. His new formula reads, “The formula with Gödel number sub(n, n, 17) cannot be proved.” Let’s call this new formula G.

 

Naturally, G has a Gödel number. What’s its value? Lo and behold, it must be sub(n, n, 17). By definition, sub(n, n, 17) is the Gödel number of the formula that results from taking the formula with Gödel number n and substituting n anywhere there’s a symbol with Gödel number 17. And G is exactly this formula! Because of the uniqueness of prime factorization, we now see that the formula G is talking about is none other than G itself.

 

Q proof?