Final clubs - a basket demonstration

["Jorge Llambias" <jjllambias@hotmail.com>]

Here is a demonstration that for final clubs to be well
defined by the circular "definition", it is necessary that
all clubs be final clubs.

Demonstration:

We will divide all clubs into two baskets,
one labeled "candidates for finality" and the other
labeled "the rest". (If the clubs are too big to fit
in the baskets, we can use little numbered balls to
represent the clubs.)

Choose any club whatsoever to start with and place it
in the "candidates for finality" basket.

Choose any of the remaining clubs and see whether it
is incompatible with the first one you chose. If it is,
put it in the "candidates" basket, else put it in the
"the rest" basket.

Choose any of the remaining clubs and test it for
compatibility with all clubs in the "candidates" basket.
If incompatible with all of them, put it there. Else
drop it with "the rest".

Repeat the preceding paragraph until there are no
clubs remaining.

Now, you can see that the "candidates for finality"
are a possible set of final clubs. They are by construction
incompatible with one another, and also by construction
every club in "the rest" is compatible with at least one
of them, so the "candidates" are a good set of final clubs.

But the catch is that I could have started the whole
procedure with any club whatsoever. So if there is
any club in "the rest", I could have started with it
and gotten a different but valid set of final clubs.

Therefore, the only way for there to be a unique possible
set of final clubs is that the "the rest" basket be empty.
This means all clubs have to be final if final clubs
are well defined. QED

Please let me know if you find any holes in my argument.

co'o mi'e xorxes


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