Aha! Halls you are right! Forgive me, but now I see for the first time, there really is no way to decide which element is "first"
.
For example when a = b, we have the proposed definition of the "ordered pair" (a,b) as the set {{a},{a,a}}. Now since {a} = {a,a}, this is just
{{a},{a}}, which of course is just {{a}}.
Thus there is no first copy of the element a.
Now this is clearly nonsense.
I.e. suppose we define an ordered singleton as a set of form {{a}}.
Then there is a non trivial intersection between the colection of ordered singletons and the collection of ordered pairs!
I.e. then when given the object {{a}}, we have no way of knowing whether it is an ordered singleton or an ordered pair.
This is obviously ridiculous and only someone who never planned to use them, would define them this way.
For instance with this definition, one could not tell the difference between the point (a,a) of the diagonal in the euclidean plane, and the point a on the real line.
So in fact the definition proposed for the ordered pair (a,b), namely {{a},{a,b}}, is a bit odd.
So unless I have made another of my many flagrant errors of thought (which is highly likely, given the huge number of intelligent people who have dutifully repeated this definition in their books), this silly definition is not only unimportant, but has odd properties.
e.g. in R^3, how should we define an ordered triple? perhaps we set (a,b,c) = the ordered pair (a,(b,c))? i.e. {{a},{{a},{b,c}}}.
Oho! We get the following: the "right" silly definition in this tradition, for an ordered singleton is of course not {{a}} but {a}.
then the ordered pair (a,a) is just {{a},{a,a}} = {{a}}.
and the ordered triple (a,a,a) is hmmm... {{a}, {{a}}}, so now R is disjoint from R^2, etc...
wheee... this kind of trivial nonsense is fun, until you wake up and realize you have more important things to do. It is kind of addictive though, like watching television. I fear it is equally harmful, as I am losing time I could be spending thinking about something deeper, like differential equations, or skew symmetric line bundles on curves.
