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Re: Lists
Posted:
Jan 24, 2013 1:40 PM
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Don Deluise wrote:
>
> This list enumerates all 2 bit binary sequences:
>
> 00
> 01
> 10
> 11
>
> This one enumerates all 3 bit binary sequences:
>
> 000
> 001
> 010
> 011
> 100
> 101
> 110
> 111
>
> Using diagonalization on the first to produce a sequence that is not in the
> list fails. It produces '10' which *is* in the list.
>
> Using diagonalization on the second list also fails. It produces '111'
> which *is* in the list.
>
> Clearly, increasing the length of the sequences does not produce lists in
> which diagonalization will achieve its purpose, i.e. to produce binary
> sequences of a given length which are not already in the list.
>
> So if we were to compile an enumeration of infinite length binary
> sequences, how do we know that diagonalization produces a sequence not
> already in the list?
Infinite things don't behave the same as finite ones.
--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
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