Re: No Unique Initial Segment And No Characteristic Expansion.

From: tinyurl.com/uh3t (rem642b_at_Yahoo.Com)
Date: 12/05/04 

Date: Sun, 05 Dec 2004 12:55:46 -0800

> From: "HERC777" <herc777@hotmail.com>
> My point is the diagonal you made, you call it unique yet it has No
> Unique Initial Segment.

Given any particular countably-infinite sequence, there is a unique
initial segment of length 1, and a unique initial segment of length 2,
etc. for any value of n you choose. Of course if you allow the length
to vary, there's no overall unique initial segment, because a segment
of length 1 is different from a segment of length 2, etc. So what does
this trivial fact have to do with the topic being discussed?? The whole
sequence may be unique in some context, and any initial segment is
unique in the context of initial segments of the whole sequence of that
particular length, but any initial segment is not unique in the context
of all initial segments of all various lengths.

> The length of all initial segments on an infinite list is unbounded.

Obviously that's true, but who cares??

> Therefore, the diagonal sequence is already present on the list

No, that's not true, and there's no valid reason why the previous
statement has any bearing on this statement, so "therefore" makes no
sense in what you wrote.

That's also irrelevant to the discussion of the anti-diagonal sequence.

> 0.3
> 0.33
> 0.333
> ...
(A countably infinite list of numbers, each of which is expressed by a
finite bunch of digits. I.e. each element on the list is a terminating
decimal fraction.)

> 0.3333...
(A single number, which is expressed by a countable-infinite sequence
of digits. I.e. this number is *not* a terminating decimal fraction.)

> Why is 0.3333.. not on that list?

Because it is not a terminating decimal fraction, whereas every element
on the list is a termianating decimal fraction. It's the same as if you
made a list of eggs, each element is an egg, nothing else ever on your
list except eggs, and you showed me a potato and asked why that potato
is not on your list. Well, because it's a potato, whereas your list has
only eggs on it.

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