Anyone can pick “the A,B,C ids” “in any wished order.” No one needs your “tree” to do or understand that. However you again specifically assert that your ‘tree’ representation for (B,A,C) is the same representation for (C,A,B)
By Standard Math {A,B,C} = {C,A,B} etc … where order has no significance.
On top of this basic state of distinct elements, one can use order.
There is an hierarchy of dependency here where some order of {A,B,C} distinct members depends on the existence of this distinction, but not vice verse.
So is the case about ON's elements order.
But ONs go deeper than distinct only elements and use Distinction in a more comprehensive way, by using n-Redundancy x n-Uncertainty tree, where set or mutiset are some particular case of it.
The Man said:
For “(2,2)” the ordering distinctions of (BA, AB), (BA, BA) and (AB, BA) are excluded or considered indistinct by your notions that you claim are primarily about distinction.
For “(2,1)” the ordering distinctions of (BA, A) and (BA, B) are excluded or considered indistinct by your notions that you claim are primarily about distinction.
(1,2) ordering distinctions like (A, AB), (B, AB), (A, BA) and (B, BA), are not even included or considered distinct by your notions that you claim are primarily about distinction.
For “(1,1)” the (B,A) ordering distinction is excluded or considered indistinct by your notions that you claim are primarily about distinction.
1) You do not understand the differentness between AB and A,B.
AB is a superposition of ids of a single element, so AB or BA is exactly the same thing (Uncertainty).
2) A,B is for two different ids of two elements, where order has no significance in that level exactly as in {A,B}={B,A} case (we can use this clear A,B ids on order to use it as the basis of some order, but then this is not at the fundamental level).
3) We also use, for example A,A (Redundancy of two elements, etc ...).
4) Here is an extension of ONs where also 0 case is considered, for example, let us use
2-Uncertainty x 2-Redundancy tree:
Code:
2X2
(AB,AB) (AB,A) (AB,B) (AB) (A,A) (B,B) (A,B) (A) (B) ()
A * * A * * A * . A * . A * * A . . A * . A * . A . . A . .
| | | | | | | | | | | | | | | | | | | |
B *_* B *_. B *_* B *_. B ._. B *_* B ._* B ._. B *_. B ._.
(2,2) = (AB,AB)
(2,1) = (AB,A),(AB,B)
(2,0)= (AB)
(1,1) = (A,A),(B,B),(A,B)
(1,0)= (A),(B)
(0,0)= ()
The beautiful thing about ONs ( as clearly shown in
http://www.internationalskeptics.com/forums/showpost.php?p=4895440&postcount=5000 ) is that all these preperties (whether they are ordered or not) a reduced into a single organic structure.
) and I own a copy of 'Ayurvedic Mathematics'.