Re: HOW MANY DIGITS OF PI HAVE PROPERTY X ?

From: george (greeneg_at_cs.unc.edu)
Date: 01/12/05 

Date: 12 Jan 2005 12:27:00 -0800

|-|erc wrote:
> Pi = <314159...............................................>
> |<------how many digits---------->|
>
>
> Let property X = the digit,

Incoherent, almost to the point of being ungrammatical.
A property CANnot EQUAL a digit. Try again,
THIS TIME IN ENGLISH.

> and every preceding digit up to that digit
> occur in order in the right spot on
> (a member of) list Y

Well, you're getting closer.
But talking about these things as "digits of pi"
is simply wrong. What they ACTUALLY are is just
digit-POSITIONS, just NATURAL NUMBERS. They could
be that position in ANY decimal expansion of ANY real.
There is nothing pi-specific about the NUMBER of places
past the decimal-point that anyone could go. When a
substring (of a string) is all the characters in that string
from the first up to some given finite number, the substring
is a PREFIX of the string and THAT is the word you need to be
using. If ANY prefix of pi occurs in ANY element of Y,
then ALL SHORTER prefixes occur in the SAME element.
So it is redundant to ask "how many" digits have this property.
If any digit (or digit-POSITION) has it, then all earlier ones
must as well. So all you really need to know is which, if any,
position is the rightmost one with this property.

>
> Let Y = {
> <333333333..>
> <300000000..>
> <399999999..>
> <314314314..>
> ..
> }
> Y has infinite members.
> The above is just a sample.
> Y is computed by UTM(row, col) mod 10
> Y includes all computable numbers for some numeric representation
>
>
> The answer should be a quantity

Not exactly. The answer will be an ordinal.
Pi ITSELF is computable, so if Y includes all computable
numbers, Pi ITSELF occurs ON THIS LIST AS A MEMBER OF Y.
THAT member gets it right for ALL digit-positions.
Note that Y is not actually computable in any practical sense;
indeed, most individual members of it, including Pi, are not
"practically" computable either, because their representations
under this paradigm are NOT FINITE or do not have standard finite
abbreviations.

> that is not related, dependant, or refers to Y.

The answer is all of them. All of them is w.

>
>
>
> Example :
> Pi = <314159.......................................................>
> |<------how many digits---------->|

All non-terminating decimal-expansions-of-a-real, EVERY real,
under-the-usual-paradigm, have w digits. When the LAST w digits
of these w digits are all 0's or all 9's, the paradigm allows a
finite abbreviation.

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