Re: A questionable existence in ZFC

From: namducnguyen (namducnguyen_at_shaw.ca)
Date: 09/09/04 

Date: Thu, 09 Sep 2004 13:41:42 GMT

David C. Ullrich wrote:

>On Thu, 09 Sep 2004 05:08:45 GMT, namducnguyen <namducnguyen@shaw.ca>
>wrote:
>
>
>
>>A decimal-string is an infinite sequence, from left to right, of characters
>>{'0'-'9'}. A Pi-sub-string is a decimal-string that would match the
>>decimal expansion of Pi: one for one, from left to right, for n positions
>>where n >=1. [It doesn't matter what character values happen to be in
>>positions n+1 and onward, as long as they are any of {'0'-'9'}]. For
>>example, given Pi = 3.14159.... then
>>
>>'1xxxx....', '14xxxx', '141xxxx', '1415xxxx...', '14159xxxx...', ....
>>
>>[where xxx is any decimal digit-character '0'-'9'.]
>>
>>are Pi-sub-strings. One way to represent a Pi-sub-string in ZFC is
>>using a countable set of ordered pair (n,m) where n is in N, and m is in
>>{0,1,...9}.
>>
>>Let's define "Pi-complete-string" to be the Pi-sub-string that completely
>>matches the entire decimal expansion of Pi.
>>
>>Let's now define a Pi-set, say, PI as:
>>
>>(1) (s \in PI) => (s is a Pi-sub-string)
>>(2) for any n >=1, there exists in PI a Pi-sub-string that matches with the
>> decimal expansion of Pi, in at least n position, from left to right.
>>
>>It doesn't seem intuitive to me whether or not Pi-complete-string should
>>be in PI.
>>
>>
>
>huh? why in the world would you think that 1 and 2 imply this?
>the set
>
>{'1000...', '14000...', '141000...', '1415000...',...}
>
>is obviously a Pi-set.
>

I thought about the un-ending 0's in each string. But is (2) enough to
exclude the membership of Pi-complete-string, having permitted those
with un-ending o's? Seems like (1) doesn't rule out Pi-complete-string,
which is a Pi-sub-string? Are we saying that (1) is not necessary to
define a Pi-set?

I'm not saying I'm correct; just that I'm not sure the effect of (1) and
(2) on each other.

>
>
>
>>What would you think? Do we need another axiom to force
>>Pi-complete-string to be in,
>>even having AC? Thanks in advance for any suggestion.
>>
>>---Nam
>>
>>
>
>
>************************
>
>David C. Ullrich
>
>sorry about the inelegant formatting - typing
>one-handed for a few weeks...
>
>

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