Re: How many digits is pi computable to?

From: Ed Murphy (emurphy42_at_socal.rr.com)
Date: 01/18/05 

Date: Tue, 18 Jan 2005 05:24:17 GMT

On Tue, 18 Jan 2005 11:59:33 +1000, |-|erc wrote:

> Infinite people each flip coins infinite times. Can you always find a
> different sequence of heads and tails?

[snip diagonal argument]

> its really quite simple, infinite people all doing the same thing you are
> doing dispells any possibility of you being unique.

"Infinite" is insufficiently precise; some are larger than others.

P = number of people
C = number of coin flips per person
S = number of possible sequences of coin flips

C is countably infinite, but S is uncountably infinite.

If P is countably infinite:
  * It can't cover all of S.
  * Your comment is false.
  * The diagonal argument works.

If P is uncountably infinite:
  * It can cover all of S.
  * Your comment can be true.
  * The diagonal argument doesn't work.

Relevant Pages

  • Re: Review of Mueckenheims book.
    ... ....2222, they are actually infinitely distant elements of a sequence, ... smallest positive number, on the infinite scale. ... That's the ordering used by Cantor to prove their countability. ... If a set S is countable, then there is a total order < of S such that ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... I have always allowed that sequences can be countably infinite, ... like the sequence of standard naturals is. ... TO nor anyone else has produced any axiom system in which it can be ... at several removes from pure mathematics, ...
    (sci.math)
  • Re: Chex Wat: Pi is "random" and "not predictable"?
    ... their output sequence. ... The fact is that an algorithm can also be used to ... first was about the probability of finding an apparent match. ... They may be matched within e at an infinite ...
    (talk.origins)
  • Re: Multiple infinities - one more look
    ... continued for lager length of digit sequences without limit. ... infinite digit sequences... ... so the resulting reals have an order. ... (i.e. having a finite program to output their digits in sequence). ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... distances that make it a curve instead of a straight line, ... In the above sequence, n is a (strictly ... the diagonal is the "infinite case" of the staircases. ... closer to 2*pi as n -> oo. ...
    (sci.math)
Quantcast