That's such a beautiful definition! It depends on your language of propositions in an interesting way: if your propositions are first-order predicate logic over the language of rings, then 'random' = algebraic. Probably you want second-order arithmetic? Reals are, a la descriptive set theory, sets of naturals. -- APG.
Sent: Wednesday, December 14, 2016 at 8:23 AM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] record computation of Pi
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
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