Normality to a given base is surely a desirable thing to know about a number. But my definition of a "random" number is as follows: Let {P_k} be the set of all propositions in about individual real numbers (otherwise with no free variables) such that for each k, the set of real numbers not satisfying P_k has measure 0. There are only countably many propositions P_k, so we may take the intersection S of all sets of numbers S_k where S_k = {x in R | P_k(x) is true} Since each one of these has full measure, the same is true of their countable intersection S. Thus S is the set of all real numbers that satisfy all the propositions satisfied by almost all reals — and almost all reals lie in S. Of course, all numbers in S are normal to every base, since almost all numbers must be that. But their lack of abnormality goes infinitely further. —Dan
On Dec 13, 2016, at 5:56 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
Plouffe's algorithm that gives the nth binary digit of pi without giving any others may lead to a proof of pi's normality in binary