Re: A simple undiagonalisable list - ILLUSTRATED


HERC777 wrote:
> i.e. the property should hold for all
> permutations of elements of the
> list.


Well, it doesn't.

SOME properties (like being what you call "saturated")
hold for all permutations of a list, but most (including
"being a contender") don't. Somebody on another thread has
also raised an important point about infinitary permutations.
These have the property that they can disturb the "connectedness"
or ordinality of the list. Initially, everything on the list
is only finitely far away from the top. If you allow infinite
permutations, then some things could become infinitely far down.
You could for example, starting at time t=1/2 = 1- (2^-1),
swap the first element of the list with the 2nd.
Then, you could at time t=3/4 = 1- (2^-2), swap the 2nd
(NOW 2nd, originally 1st) element with the 3rd.
And at t=7/8 you could swap the 3rd (originally 1st) element
with the 4th. If you do this "all the way" (infinitely many
times) until t=1, then you get a "list" of order-type
w+1 where the last (w-th or oo-th) element comes after
the infinitely many elements 2,3,... of the original list.
By the usual definitions
this is NOT even a "list" at all... it is ESPECIALLY not a
"countable list" because, UNlike the set of counting/natural
numbers, IT HAS A LAST ELEMENT.

.

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