Hal, I think you really might want to read some introductory textbook on logic and formal systems, to check the standard definitions of `proof' and `theorem.'
BTW, the following remarkable method heavily depends on what's provable. I believe it will find its way into general computer science textbooks. ------------------------------------------------------------------------- The fastest and shortest algorithm for all well-defined problems Marcus Hutter, IDSIA An algorithm M is described that solves any well-defined problem p as quickly as the fastest algorithm computing a solution to p, save for a factor of 5 and low-order additive terms. M optimally distributes resources between the execution of provably correct p-solving programs and an enumeration of all proofs, including relevant proofs of program correctness and of time bounds on program runtimes. M avoids Blum's speed-up theorem by ignoring programs without correctness proof. M has broader applicability and can be faster than Levin's universal search, the fastest method for inverting functions save for a large multiplicative constant. An extension of Kolmogorov complexity and two novel natural measures of function complexity are used to show that the most efficient program computing some function f is also among the shortest programs provably computing f. ftp://ftp.idsia.ch/pub/techrep/IDSIA-16-00.ps.gz ------------------------------------------------------------------------- Juergen Schmidhuber www.idsia.ch
