Infinite number of infinite coin flips

From: Bill Smythe (chichess_at_beforeRCNafter.com)
Date: 01/19/05 

Date: Wed, 19 Jan 2005 10:46:48 -0600

"Ed Murphy" wrote:
> To summarize:
>
> P = number of people
> C = number of coin flips per person
>
> If P is countably infinite and C is countably infinite, then the
> diagonal argument works.
>
> If P is countably infinite and C is uncountably infinite, then the
> diagonal argument still works. (Consider a countably infinite
> subset of each person's sequence of coin flips. This case now
> reduces to the previous case.)
>
> If P is uncountably infinite, then the diagonal argument doesn't
> work. (It relies on a bijection from the set of people to the
> natural numbers; if P is uncountably infinite, then by definition
> there isn't one.)

What if P and C are both uncountably infinite (say, to the same degree of
aleph-whatever)? Is there a variation of the diagonal argument which will
work in this case?

> And now let's set aside all these side issues that may or may not
> pertain, and revisit your original question. "How many digits is
> pi computable to?" ....

You're right, we should have changed the thread title a long time ago -- as
I have now done.

> .... Or, to avoid digressions into practical physics,
> and also to explicitly address the difference between countable and
> uncountable infinities: "Does the complete decimal expansion of pi
> consist of a countably or uncountably infinite number of digits?"

Huh?? The first digit after the decimal point is number 1. The next is
number 2. Etc. There's your bijection.

Bill Smythe

Relevant Pages

  • Re: How many digits is pi computable to?
    ... >> the same as my first X flips. ... I have been assuming "a countably infinite number". ... whether it's meaningful for this number to be uncountably infinite. ...
    (comp.theory)
  • Re: How many digits is pi computable to?
    ... >> the same as my first X flips. ... I have been assuming "a countably infinite number". ... whether it's meaningful for this number to be uncountably infinite. ...
    (sci.math)
  • Re: How many digits is pi computable to?
    ... >> the same as my first X flips. ... I have been assuming "a countably infinite number". ... whether it's meaningful for this number to be uncountably infinite. ...
    (sci.logic)
  • Re: Cantor and the binary tree
    ... to assume that one of these numbers becomes uncountably infinite while the other remains countably infinite. ... One can hardly imagine a simpler mathematical proof. ...
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  • Re: Well Ordering the Reals
    ... You are confusing arithmetic exponentiation and the cardinality of powersets - they are not the same thing, ... These are precisely the types of expressions I want to see distinguished, not all lumped together as if arithmetic becomes meaningless for infinite values. ... Since each term represents the number of points your mapping defines at each step, we conclude that your mapping defines only a countably infinite number of points in total. ... a countably infinite number of points, which is far too small to cover the reals. ...
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