by **DonRetrasado** » Tue Jul 19, 2011 4:32 am

Kaharz wrote:DonRetrasado wrote:This is all well and good and I believed that such a thing could exist. What I am more interested in is an actual mathematical proof that explains the "statistical certainty" of finding any string of words.

All possible elements exist in any infinite set. Since pi is theoretically an infinite set, all elements exist within it. Since it is theoretically not ordered in any way, all possible subsets of positive integers exist in it as well. This is of course provided that all digits 0 through 9 occur infinitely in pi. Everyone is pretty sure that pi is irrational (non-terminating, non-repeating), but whether or not it is a normal number is unproven.

IIRC, to have a 100% certainty that Hooray for Zorblax would appear somewhere in pi, you would have to scan 26^16 digits, or about 4.36x10^22 digits. From a quick google search, it looks like the current record for digits of pi calculated is only 5 trillion, or 5x10^12.

I believe you, it's just I'm not quite wrapping my head around it. How do we know that all possible subsets exist? (ex. if not all digits occurred infinitely, it would need be true that a string of words exists such that it would not exist in this number)

It seems to me that there is a possibility that, because of the unknown nature of pi's set, certain strings of letters/numbers might simply not exist, but I'm not 100% confident of this either.

[quote="Kaharz"][quote="DonRetrasado"]This is all well and good and I believed that such a thing could exist. What I am more interested in is an actual mathematical proof that explains the "statistical certainty" of finding any string of words.[/quote]

All possible elements exist in any infinite set. Since pi is theoretically an infinite set, all elements exist within it. Since it is theoretically not ordered in any way, all possible subsets of positive integers exist in it as well. This is of course provided that all digits 0 through 9 occur infinitely in pi. Everyone is pretty sure that pi is irrational (non-terminating, non-repeating), but whether or not it is a normal number is unproven.

IIRC, to have a 100% certainty that Hooray for Zorblax would appear somewhere in pi, you would have to scan 26^16 digits, or about 4.36x10^22 digits. From a quick google search, it looks like the current record for digits of pi calculated is only 5 trillion, or 5x10^12.[/quote]

I believe you, it's just I'm not quite wrapping my head around it. How do we know that all possible subsets exist? (ex. if not all digits occurred infinitely, it would need be true that a string of words exists such that it would not exist in this number)

It seems to me that there is a possibility that, because of the unknown nature of pi's set, certain strings of letters/numbers might simply not exist, but I'm not 100% confident of this either.