Understanding Ordered Pairs to Notation and Definitions

[Hurkyl]

I've always wondered about the existence 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')

 

[Sir_Deenicus]

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]

note however the difference between this and the more usual definition of a set of ordered pairs as a map with domain {1,2}. then the ordered pair (a,b) is a function f with f(1) = a and f(2) = b, so there are two "copies" of a, one taken first and one taken second.
to say that when both elements of the pair are equal, one does not need to designate one as first is different from saying there are two elements, one first one second, and both are equal. that is all i ma saying: i n ever noticed thius distinction before, and it violates my sense of...
hey what am I doing! I am going on with this trivial topioc. also if you read the remainder of my previous post you will see i have already pointed out the answer to my own paradox.
notice that in most cases a math talk can never be too elementary, the more elementary it is the mroe people understand and the mroe they respond. similarly a thread can never be too trivial. the moreso, the mroe resposnes it generates,
perhaps if i ask why 1+1 = 2, we will have another deathless thread.