I'm not sure if I completely understand your point. Is it that the definitions of ordered pairs must be done carefully when talking about constructions in Principia because of its formulation in logical predicates, e.g. care was taken when constructing sets to avoid Russell's paradox explicitly given the axioms of logic rather than Russell's paradox being excluded in ZF by the axiom schema of specification?
Or is the difficulty in introducing a canonical order for the ordered pair, or introducing well/partial-ordering in sets themselves? I guess I see an ordered pair as more of an indexical definition than an ordering definition.
As the Principia is pretty ugly and tortured at least for me let me offer:
Naive set theory. Halmos, Paul R. http://people.whitman.edu/~guichard/260/halmos__naive_set_th...
Note the first entry of "Ordered Pairs"
> What does it mean to arrange the elements of a set A in some order?
Also note how the earlier section on "Unordered Pairs" is more about building the axiom of pairing etc...to get to ordered pairs which gets to the Cartesian product, which outputs ordered pairs.
It doesn't matter if you go through Zermelo's theorem+Zorn, that states that every set can be well-ordered, or though Cartesian product's and/or AC. (Note: This is in FoL well-ordering and AC are the same, but not in SoL and HoL)
It is not that sets are expressly unordered, as a set of points in a line segment would very much have an order, but that you didn't actively arrange the elements in order to take advantage of properties that are useful to you.
Maybe I just hit mental blocks but IMHO it is important that when you make the assumption that "there exists a set." it is very important to realize that it is "unordered" because you haven't imposed one, but is not an innate property of an element of the set.
Hopefully that helps in addressing this from your original post.
> "ordered pair" is not part of set theory
While many creators of both naive and formal set theories may choose to not define (a,b) = {{a},{a,b}} explicitly, the output of the Cartesian product is the ordered pairs, so it doesn't matter, you don't have a useful set theory without them.