For it is a 'fact of experience' or so at least we are sometimes told that chance like empirical sequences show that peculiar I still consider my old doubt concerning the assumption of an axiom of convergence, and the possibility of doing without it, perfectly justified: it is justified by the developments indicated in appendix iv, note and in appendix vi, where it is shown that randomness (if defined by 'shortest random like sequences') entails convergence which therefore need not be separately postulated.