my forty years on his shoulders - Department of Mathematics

25remarks by R. Parikh, it is likely that Gödel had inadvertently used lengths, andprobably intended Gödel numbers or numbers of symbols.In any case, the analogous result with Gödel numbers was proved in(Mostowski 1952). Similar results were also proved in (Ehrenfeucht, Mycielski1971) and (Parikh 1971). Also see (Parikh 1973) for results going in the oppositedirection concerning the number of lines in proofs in certain systems.In (Friedman 79), we considered, for any reasonable system T, andpositive integer n, the finite consistency statement Con n (T) expressing that “everyinconsistency in T uses at least n symbols”. We gave a lower bound of n 1/4 on thenumber of symbols required to prove in Con n (T) in T, provided n is sufficientlylarge. A more careful version of the argument gives the lower bound of n 1/2 forsufficiently large n. We called this “finite second incompleteness”.A much more careful analysis of finite second incompleteness is in(Pudlak 1985), which establishes an (n(log(n)) -1/2 ) lower bound and an O(n)upper bound, for systems T satisfying certain reasonable conditions.It would be very interesting to extend finite second incompleteness inseveral directions. One direction is to give a treatment of a good lower bound fora proof of Con n (T) in T, which is along the lines of the Hilbert Bernaysderivability conditions, adapted carefully for finite second incompleteness. Weoffer our treatment of the derivability conditions in section 4 above as alaunching point. A number of issues arise as to the best way to set this up, andwhat level of generality is appropriate.Another direction to take finite second incompleteness is to give someversions which are not asymptotic. I.e., they involve specific numbers of symbolsthat are argued to be related to actual mathematical practice.