Why do you regard those few naturals as an infinite incomplete collection?
Because you used ", ..." at the end of some given natural numbers, which means that the given set is infinite.
Since it is infinite it is also incomplete, according to my axiom.
I didn't say that the infinite collection was ordered; that the naturals were organized in the ascending order.
Order has no significance among the members of a given set (where a set is a collection of distinct objects).
The standard definition of a collection of items called the set allows the members of the set be organized in no particular order. Your definition of the set is very likely different.
Wrong, it uses the same notion about the insignificance of order among distinct members of a given set.
If you look at the diagram in
http://www.internationalskeptics.com/forums/showpost.php?p=7601080&postcount=16418 you can see that the points are not picked in some particular order, whether the collection of distinct points is finite, or not.
EDIT:
Again.
As for no order, for example, (AB,AB)
really has no order (which is not the same as "order has no significance") exactly because the considered framework is under superposition of identities of 2-Uncertainty x 2-Redundancy Distinction Tree.
According to this example (without a loss of generality) sets are based on Distinction State (A,B) = (B,A) under F(1,1).
Please be aware of the fact that
(A,B) is just a particular case of Frame
(1,1) under 2-Redundancy x 2-Uncertainty Distinction Tree, as follows:
Code:
(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)
[COLOR="magenta"][B](1,1)[/B][/COLOR] = (A,A),(B,B), [COLOR="magenta"][B](A,B)[/B][/COLOR]
(1,0)= (A),(B)
(0,0)= ()
Since the members of {A,B} are based on the particular case of DS (A,B) under F (1,1), and since {A,B} has 0-Uncertainy and 0-Redundancy, then its members are pick-able no matter if {A,B} = {B,A} (or in other words, order has no significance).
C is a set (where
set is a collection of distinct objects, which is not empty if it has pick-able objects).
Here is my axiom:
If (x in C is picked) AND (everything but x, in C is picked) AND (x can't be picked twice), then C is infinite AND incomplete.
As can be seen, no particular order is involved here.
Again, more about order:
Let's assume that a given collection of distinct objects (known also as set) has no order.
In that case non of its objects can be picked and used, because any attempt to pick something from a given collection of distinct objects, must be the first pick, and if there is the first pick, then there is the second pick, etc ...
In other words, if order does not exist among collections of distinct objects, then their distinct objects are not available.
For example, the expression "647+23" does not exist, since the objects of the collection of natural numbers are not available.
So, "No order" is not the same as "Order has no significance".
