Re: Easy question: WHICH LIST CONTAINS MORE DIGITS OF Pi?

From: Ed Murphy (emurphy42_at_socal.rr.com)
Date: 02/09/05 

Date: Wed, 09 Feb 2005 06:17:13 GMT

On Wed, 09 Feb 2005 15:04:17 +1000, |-|erc wrote:

> "The Ghost In The Machine" <ewill@sirius.athghost7038suus.net> wrote
>> >> The list {3, 3.1, 3.14, ...} contains all finite prefixes of the
>> >> decimal expansion of pi. However all elements of this set have finite
>> >> length, thus the "full" decimal expansion of pi is not a member of
>> >> the set, by which I mean pi is not a member of the set.
>> >>
>> >> As far as I can make out, this arguement started when Herc asked a
>> >> question informally, and george complained about having to assume a
>> >> fairly clear interpretation of herc's question. maybe this battle
>> >> would be better fought out in the thread entitled
>> >> Name the thesis: "Formal sentences capture informal ones"
>> >>
>> >> May the best man win.
>> >>
>> >> Joe
>> >
>> > the motivation is the fact no one in sci.math can answer simple
>> > questions.
>> >
>> > How many digits of this infinite sequence <2983738...> have property
>> > X?
>>
>> All finite prefixes have property X.
>
> that is fucking idiotic! read the question for fucks sake

Read the answers. When twenty people tell you that you're being
ambiguous, it's time to give serious consideration to the idea that
they're right. Especially when you've been told repeatedly that
the difference between countably-infinite and uncountably-infinite
is crucial, and yet you keep glossing over it. This is *not*
orbital mind control lasers, it's just you being dense or trollish
or both.

> how many digitos of <123...> are numbers?

Here is an unambiguous question (at least I think it's unambiguous),
which may or may not be what you were trying to ask above (I cannot
read your so-called mind):

"Let S1 be a countably infinite sequence <123...> of digits, each of
 which is a member of the set {0,1,2,3,4,5,6,7,8,9}. Let f() be such
 that f(1) = 0.1, f(2) = 0.12, f(3) = 0.123, and in general
 f(n) = sum (x = 1 to n) (x'th digit of S1 / 10^x). Let S2 be the set
 of all integers for which f(x) is rational. What is the cardinality
 of S2?"

Answer: Countably infinite.

And here is a follow-up question:

"Consider the specific case in which S1 consists of the digits of
 pi. Does the countably infinite cardinality of S2 imply that pi
 is rational?"

Answer: No.

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