Re: No Unique Initial Segment And No Characteristic Expansion

From: |-|erc (h_at_r.c)
Date: 12/10/04 

Date: Sat, 11 Dec 2004 09:28:03 +1000

"George Greene" <greeneg@quartet.cs.unc.edu> wrote in
> "|-|erc" <h@r.c> writes:
>
> : "Kenneth Doyle" <nobody@notmail.com> wrote in
> : > "|-|erc" <h@r.c> wrote in news:31oaloF3e8elfU1@individual.net:
> : >
> : >
> : > > DMC's diagonalizer is never
> : > > unique,
> : >
> : > His list is guaranteed to be a mismatch with the next person's list; at the
> : > same ordinal position on both lists. That holds true every time the
> : > diagonalizer visits the next person. How could this be any simpler?
> : >
> :
> : How could this be any simpler.
> :
> : H
> : T
> : HH
> : HT
> : TH
> : TT
> : HHH
> : HHT
> : HTH
> : HTT
> : THH
> : THT
> : TTH
> : TTT
> :
> : all these prefixes have been flipped.
>
> Absolutely ALL of THESE prefixes ARE *FINITE*, DIP***.
> NO entry on this list EVER has ANY INfinite PREfix!

Its clearly labelled a list of prefixes, hence a list of finite objects.

>
> : the length to which they have all been flipped for, is infinite.
>
> No, each FINISHED line on the list is infinite, but every PREFIX
> of every line on the list is finite.

the length of prefixes approaches infinity. this is overlooked, this is my only point,
it negates the fact that changing digits makes unique sequences.

>
> : oo options for the diag
>
> No, there is only ONE option for the diag if there is
> only one list and it is in binary.
>
> : - oo different sequences on the list
>
> There are the SMALLEST infinity of different sequences on
> the list. But there is a much BIGGER infinity of possible
> LISTS. The diag can be one of THAT infinitely bigger number.
>
> There are MORE infinite sequences of HT *beginning* with each
> of your prefixes above THAN there are prefixes!
> The number of finite prefixes (or sequences) is countably
> infinite but the number of denumerably-long ones is a bigger
> infinity than that.

There are more infinite sequences beginning with each prefix than there are prefixes.
as in
There are uncountable infinite sequences beginning with countable infinite prefixes.

#sequences > #prefixes

yet, each sequence has multiple prefixes, for any finite number of sequences
#prefixes > # sequences

CONTRADICTION
or
Why does this inequality change sides as the number of sequences -> oo?

Herc 

--
> then it is uniquely defined AFTER the flippers have flipped
> enough (given that every flipper is only a finite number of
> flippers from the beginning, and every flip is only a finite
> number of flips from the beginning, no individual flip or anti-
> flip needs infinite inputs to compute).   GEORGE GREENE sci.logic
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