YOU IGNORAMUSES ACTUAL:LY BELIEVE THIS ?

From: |-|erc (H_at_r.c)
Date: 01/26/05 

Date: Wed, 26 Jan 2005 13:06:29 +1000

[Herc]
>>>Here's a sequence
>>>
>>><1234567898765432>
>>>
>>>Here's a list
>>>
>>><111111111111111>
>>><121212121212121>
>>><123123123123123>
>>>
>>>
>>>Now put a bar after the 1st digit position
>>>
>>> |
>>><1 | 234567898765432>
>>> |
>>><1 | 11111111111111>
>>><1 | 21212121212121>
>>><1 | 23123123123123>
>>> |
>>>
>>>
>>>How FAR can you move the BAR?
>>>
>>> |
>>><12 | 34567898765432>
>>> |
>>><11 | 1111111111111>
>>><12 | 1212121212121>
>>><12 | 3123123123123>
>>> |
>>>
>>>THAT PORTION OF THE SEQUENCE IS ON THE LIST TOO.
>>>
>>>Can you get any *more* than that finite amount?
>>>
>>>
>>>
>>> |
>>><123 | 4567898765432>
>>> |
>>><111 | 111111111111>
>>><121 | 212121212121>
>>><123 | 123123123123>
>>> |
>>>
>>>
>>>The SEQUENCE is on the list to 3 DIGITS.
>>

[Will]
>>Since the bottom three is supposed to be all computable
>>numbers, and all finite sequences are computable, all finite prefixes
>>are on the list of computables and you can move the bar any finite
>>distance to the right. The bar will NOT, however, reach infinitely to
>>the right.
>
>Care to prove that ?

[Will]
Construct a sequence, M, of moves, where the bar moves to the right one
space each successive move, and at the first move, M_1, it is one to the
right.

Claim: At any point M_n in the sequence, the bar has only moved a finite
distance to the right along the digits.

Proof by induction:
Base case: n=1 -> M_1 = 1 to the right, and therefore a finite distance
to the right.
Inductive case: suppose for n, M_n has moved a finite distance to the
right. Then it has moved to the right m digits, where m is in N. Then
for n+1, M_(n+1) has moved to the right m+1 digits. Since m is in N, so
is m+1 and therefor m+1 is finite and M_(n+1) has moved to the right a
finite number of digits.

Therefore: for all n in N, M_n has moved to the right a finite number of
digits.

Now, I've obviously made some assumptions in my interpretation of the
problem to show that you never reach infinitely to the right. Are there
any of those assumptions you'd care to challenge? 

--
Will Twentyman
sci.math has been overtaken by the kindergarden.  Can you move the bar over oo digits?
Herc
--
In 100 years cardinality and incompleteness theory will only be offered under modern history as cult beliefs
Veni, vidi, vamos.

Relevant Pages

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    ... >>are on the list of computables and you can move the bar any finite ... The bar will NOT, however, reach infinitely to ... At any point M_n in the sequence, the bar has only moved a finite ... Then it has moved to the right m digits, where m is in N. Then ...
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  • Re: just 5 quick answers then I can summarise and GO
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  • Re: How far can you move the bar MONOSPACED
    ... > How FAR can you move the BAR? ... > THAT PORTION OF THE SEQUENCE IS ON THE LIST TOO. ... And each time you will have moved the bar a finite number of digits to ... are on the list of computables and you can move the bar any finite ...
    (sci.math)
  • Re: How far can you move the bar MONOSPACED
    ... > How FAR can you move the BAR? ... > THAT PORTION OF THE SEQUENCE IS ON THE LIST TOO. ... And each time you will have moved the bar a finite number of digits to ... are on the list of computables and you can move the bar any finite ...
    (sci.logic)