Re: How many digits is pi computable to?
From: |-|erc (h_at_r.c)
Date: 01/18/05
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Date: Tue, 18 Jan 2005 20:43:03 +1000
"Ed Murphy" <emurphy42@socal.rr.com> wrote in ...
> On Tue, 18 Jan 2005 19:12:59 +1000, |-|erc wrote:
>
> > "Ed Murphy" <emurphy42@socal.rr.com> wrote in ...
> >> On Tue, 18 Jan 2005 15:32:40 +1000, |-|erc wrote:
> >>
> >> > Say you have an (countable) infinite set of people, and they only toss
> >> > coins a finite number of times.
> >> >
> >> > <<sample>>
> >> > P C
> >> > 1 HTHT
> >> > 2 HHTT
> >> > 3 TTHH
> >> > 4 TT
> >> > 5 H
> >> > 6 T
> >> > ...
> >> >
> >> > they are given the constraint to only toss 4 times maximum.
> >> >
> >> > you can construct any sequence you want once I show you the list, but
> >> > first you have to tell me how long your sequence is going to be?
> >>
> >> Oh, you're not giving *me* the 4-tosses-maximum constraint? Fine, then
> >> my sequence will be 5 tosses. My sequence is HHHHH. None of your
> >> people got *that* sequence, did they?
> >>
> >> Perhaps the orbital mind control lasers are interfering with your
> >> ability to say what you mean in a precise fashion.
> >
> > that's fine, note that when competing against infinite other people you
> > had to break their contraint.
>
> Ah, now I see what you're attempting to stumble toward.
>
> If the maximum length of a sequence of coin flips is finite, then the
> number of possible sequences is also finite, and an infinite number of
> people can choose them all.
>
> However, if the maximum length of a sequence of coin flips is countably
> infinite, then the number of possible sequences is *uncountably*
> infinite. At this point, it is no longer adequate to refer simply to
> an "infinite" number of people; we must specify either "countably
> infinite" (in which case they cannot choose all possible sequences)
> or "uncountably infinite" (in which case they can).
>
> Here is a reiteration of the diagonal argument, which shows how to
> construct a sequence missed by a countably infinite number of people.
>
> If a set is countably infinite, then there is a function that maps
> each element of that set to exactly one natural number. If the number
> of people (other than me) is countably infinite, and the number of coin
> flips per person is countably infinite, then the following functions
> exist:
>
> * A function f(P) that maps each person P (other than me) to exactly
> one natural number.
>
> * For every person P (other than me), a function g_P(C) that maps each
> of their coin flips C to exactly one natural number.
>
> * A function g_M(C) that maps each of my coin flips C to exactly one
> natural number.
>
> Let f'() be the inverse of f(), g_P'() be the inverse of g_P(), and
> g_M'() be the inverse of g_M().
>
> I construct my sequence as follows:
>
> For every natural number N,
> find the Nth person P = f'(N),
> find their Nth coin flip g_P'(N),
> and set my Nth coin flip g_M'(N) opposite.
>
> If any person P (other than me) chose the same sequence as me, then
> let N = f(P); but then their Nth coin is different from my Nth coin,
> contradiction. Thus no person P (other than me) chose the same
> sequence as me. QED.
>
How many flips of your 'new' coin sequence have other people done from flip 1 to flip X?
Herc
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