Order in Set theory and Logic In ZFC ordered pairs are often defined in terms of Kuratowski pairs $(x, y)=\{\{x\}, \{x, y\}\}$ or some other such construction to avoid introducing additional primitive notions. The definition is then usually followed by apologies because it implies artificial properties like $\{x\}\in (x, y)$ etc. which are then immediately ignored. This then leads to even weirder and more arbitrary relations down the line. Meanwhile the notion of an ordered strings is taken granted in the underlying logic e.g. $x\rightarrow y$. 
 Since strings are automatically equipped with order why not drop the requirement that order pairs are sets altogether and instead try to arrange something like  $(x, y)=xy$ for sets $x$ and $y$? (If we can use strings of length 1 to represent sets why not strings of length 2 to represent ordered pairs?) $(x, y)$ would then presumably be an urelement but not a set. This would have the advantage that all objects are constructed from sets even though not all are sets.
 A: Interesting question. A case could be made that set theory presupposes logic and that logic presupposes some notion of order (in addition to strings in a formal language, in a proof later statements depend on earlier statements). At best, this would show that Kuratowski's construction is not the origin of the notion of order.
Note that logic also presupposes a notion of sets (e.g. a theory is a set of statements). One of the goals of a formal set theory is to provide a rigorous foundation for such presuppositions. A formal theory of sets linked with an informal theory of order wouldn't provide an adequate foundation. The only real choice is between having a formal theory of sets such as we now have with ZFC and a formal theory of sets + order (something like ZFC extended with new primitives and new axioms). Since Kurtaowski's construction shows that we can get by with the former, the latter would just make the theory more complicated with no real benefit. 
A: A language that has no limitations on sentential self-reference is inherently inconsistent. (E.g. Sentence A says "B is false" and sentence B says "A is true".) In the set-theoretical systems ZF (Zermelo-Fraenkel) and ZFC, the limitation is 0. Strings of symbols in the formal language cannot be interpreted as assertions about other strings. Certain strings are called formulas and certain formulas are called sentences. Certain sentences are called axioms. Theorems are sentences derivable from the axioms by certain rules. We may introduce new symbols that are abbreviations for strings,  provided we do not introduce new axioms. (Unless that is our intent.)
$z=\{x,\{x,y\}\}$ is an abbreviation for the string $\forall u\;(u\in z\iff (u=x\lor [\forall s\ ;(s\in u\iff (s=x\lor s=y))])).$ From the axiom system ZF we can derive the theorem $\forall x,y \exists! z\;(z=\{x,y\}).$ And we can then insert the symbol $(x,y)$ for  $\{x,\{x,y\}\}$ wherever it appears in a formula.
