It's not about unpredictability, it's about the stronger notion of incompressibility. You can show that Chaitin's constant is algorithmically incompressible*, which implies normal. * proof: if not, you could create a program of N bits which knows how many programs (other than itself) of length <= N bits terminate, and then (by dovetailing) algorithmically determine *which* programs terminate. Then, it can decide to terminate if and only if it doesn't terminate. Best wishes, Adam P. Goucher
Sent: Thursday, December 06, 2018 at 8:33 PM From: "Andy Latto" <andy.latto@pobox.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Messages in pi
On Thu, Dec 6, 2018 at 10:02 AM Mike Stay <metaweta@gmail.com> wrote:
It's normal to base 10. I think the claim is that nobody knows a specific number to be normal to every base. That said, the claim has to be restricted to computable numbers, since an algorithmically random real like the halting probability of a prefix-free universal Turing machine has to be normal to every base; if not, you could predict infinitely many (not necessarily contiguous) digits of it, which contradicts the definition of algorithmically random.
I don't follow this. If I tell you that X has no 3s in it's base-10 expansion, which means it is not normal, how does this let you predict infinitely many digits (or even one digit!) of X?
Andy Latto
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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