Re: No Unique Initial Segment And No Characteristic Expansion.

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

Date: Sat, 04 Dec 2004 14:53:06 -0800

> From: "HERC777" <herc777@hotmail.com>
H> Infinite people each flip coins infinite times.

R>Infinite simply means not finite, which we know also means larger than
R>any finite. But there may be lots of different sizes of
R>larger-than-finite, so which particular one did you have in mind for
R>the number of people and the number of coin flips per person?

H> infinite flippers list ... and the coin flips for each person are
H> referred to as sequences. both list and sequence imply countable
H> infinity

Are you saying your original statement was supposed to mean "countably
infinite number or people" each flip coin(s) "countable infinite number
of times"? (I guess it doesn't matter whether an individual person
flips just one coin a countable infinite number of times, or switches
coins every so often when the old coin wears out, so long as the total
number of flips per person is countable infinite).

H> Can you always find a different sequence of heads and tails?

Do you mean if somehow you could collect all the h/t sequences from all
the corresponding people, i.e. build a complete 2-d array, let's call
it F, where both row index and column index run forever in one
direction, can you find a single h/t sequence that is *not* exactly the
same as any particular row in your array? Well of course! If the array
is indexed from 1,2,... forever along each axis, you can define a
single h/t sequence indexed along one axis as follows:
For any m = 1,2,..., let A(m) be the opposite of F(m,m), that is A(M)
is H if F(m,m) is T and vice versa.
This is Cantor's diagonal method, which has been the topic of several
threads you started recently. Why do you even need to ask the question
unless you were sleeping through all your threads?

> How on Earth can you exhaust an infinite set?

You know as much as anyone else the Earth is finite in size, finite in
number of particles, finite in memory capacity, etc., so of course if
you mean literally "on Earth" you cannot exhaust an infinite set.

> Aren't all possible combinations of heads and tails for infinite
> flips already been done?

It depends on what sequences of h/t you are possible. If you have coins
that aren't random, that for some perverse reason always go into a loop
where they repeat the same sequence as before, so every sequence with a
single coin generates a periodic sequence, then indeed there are only a
countable number of possible sequences, and it's possible that the
countable sequence of different coins might exactly cover all such
repeatable sequences, and then Cantor's diagonal method produces a
non-repeating sequence that is not a possible outcome of any single
person flipping a single coin, so it does produce a sequence of h/t but
such h/t sequence cannot be flipped exactly by any coin.

Now it's actually pretty easy to make a list of all repeating patterns.
Start with the two patterns of length 1:
HHHHHHHH...
TTTTTTTT...
Then do the two additional patterns of length 2:
HTHTHTHT...
THTHTHTH...
Then do the six additional patterns of length 3:
HHTHHTHHT...
HTTHTTHTT...
HTHHTHHTH...
THHTHHTHH...
THTTHTTHT...
TTHTTHTTH...
etc.
So using Cantor's diagonal on that sequence of sequences you can
directly construct a non-repeating sequence. (Note it's even easier if
you allow duplicates among the original sequences. Then you don't have
to carefully omit any large pattern that is composed of smaller
patterns, and it's easier to directly compute the mth generator and
hence the mth digit of the mth generator hence the opposite of that
mth-of-mth hence the mth of Cantor non-repeating sequence.)

Back to the countable infinite sequence of people each flipping a coin
a countable infinte sequence of times:
> With probability 1, this list contains every initial segment possible
> of heads and tails sequences.

Assuming the coins are fair, that's an understatement. Any particular
initial pattern is expected to occur an infinite number of times among
the sequence of H/T sequences.

> Characteristic Expansion

I'm not familiar with that term, and a Google search turns up that term
only in biochemistry/genetics, not in mathematics. Is it some term you
invented yourself, or can you show me somewhere on the Web where it's
defined as a mathematics term?

> No Unique Initial Segment

I'm not sure what you mean by that phrase, except in the trivial sense
that there is more than one initial segment of the h/t sequences, i.e.
not all h/t sequences start out exactly the same way. If you mean
anything less trivial than that, please define what you mean.

(Note typo: "number or people" should read "number of people", sorry,
didn't see until mid-upload.)

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