Re: WELL WHICH IS IT... ?
From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 01/21/05
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Date: Fri, 21 Jan 2005 05:01:45 GMT
In sci.logic, Will Twentyman
<wtwentyman@read.my.sig>
wrote
on Thu, 20 Jan 2005 22:20:37 -0500
<41f0754c$1_1@newsfeed.slurp.net>:
> |-|erc wrote:
>
>> > "If you have the list of computables, a random real number can
>> > be on it to an infinite number of digits, and yet not be on
>> > the list" True / False / Other
>>
>> As stated: false. The real either is or is not on the list,
>> but not both.
>>
>> ---------------------------------------
>>
>> > "If you have the list of computables, a random real number can
>> > be on it to an infinite number of digits, and yet not be on
>> > the list" True / False / Other
>>
>> True as phrased. (Example: S_3 and 1/3, TX_10 and any q whose
>> denominator has a prime factor other than 2 or 5)
>
> So, apparently Ghost and I interpretted the above differently. I
> interpretted it as "Given the list of computables, and a real number X,
> X can have all its digits as a single entry on the list, but not be on
> the list."
>
> With that reading, it is equivalent to "Given the list of computables,
> and a real number X, X can be on the list, but not be on the list,"
> which is obviously false.
>
> I'm not sure how Ghost read it.
>
Hmm....well, if one takes my favorite class TX_10, then it
is clear that all of pi's finite prefixes are within TX_10,
but pi, being irrational, is not.
Or, one can take a constant such as Chaitin's Omega.
This constant is a perfectly good real number, but is
not computable, therefore cannot be on any list of computables.
However, all of its prefixes easily can be (since one way
of "computing" a finite sequence is simply to recite it).
That was more or less my thinking; apologies if that wasn't
horribly clear.
Of course |-|erc isn't being horribly clear here, either.
Having a real number be on a list of numbers "to an infinite
number of digits" isn't all that well-defined an operation.
-- #191, ewill3@earthlink.net It's still legal to go .sigless.
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