# Ordered pair

Hurkyl
Staff Emeritus
Gold Member
I've always wondered about the existance of strange and unusual statements one might be able to prove through these standard models. For example:

3 = 1 U (0, 1)

Because:
0 = {}
1 = {0}
2 = {0, 1}
(0, 1) = {{0}, {0, 1}} = {1, 2}
3 = {0, 1, 2}
1 U (0, 1) = {0} U {1, 2} = {0, 1, 2} = 3

At one time, I satisfied myself that none of this is a worry, if you rigorously type' things. (I.E. don't try to ask about the equality of things of different types')

mathwonk said:
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.
I think the definition works, and I do not think it is meaningful to infer backwards. I think <x,y> => {{x}, {x,y}}. That when x = y, we have a singleton means that since all parameters are equal the choice of first is ambigious or trivial. Because {{x},{x,y}} is a defintion and a convention its meaning can only be inferred by context (for example with knowledge of what space you are working on). {x} is drawn out first only to state which element is first in purely set theoretic terms in order to define some concept of order.

mathwonk