Goedel incompleteness means there is no single algorithm for proving all provable theorems.
Are you sure you meant to say that? There is such an algorithm: enumerate all proofs. What doesn’t exist is an algorithm for finding a proof if one exists and reporting the nonexistence of a proof if it doesn’t.
According to Wikipedia, “an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing output and terminating at a final ending state.” There is a linked quote of Knuth who says “A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a ‘computational method’”.
Also, even if the terminology is atypical the charitable reader should be able to understand the intent.
Are you sure you meant to say that? There is such an algorithm: enumerate all proofs. What doesn’t exist is an algorithm for finding a proof if one exists and reporting the nonexistence of a proof if it doesn’t.
By “algorithm” I meant a program that terminates on all inputs. Maybe I should have been more specific.
That is very atypical terminology.
According to Wikipedia, “an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing output and terminating at a final ending state.” There is a linked quote of Knuth who says “A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a ‘computational method’”.
Also, even if the terminology is atypical the charitable reader should be able to understand the intent.