# Does there exist a pairing function which preserves ordering?

Suppose we define a total order on pairs of integers where $$(x_1, y_1) > (x_2, y_2)$$ when $$x_1 > x_2 \lor (x_1 = x_2 \land y_1 > y_2)$$, i.e., first comparing the first element and then falling back to the second when the first elements are equal.

Does there exist a pairing function which preserves this ordering?

Consider the pairs $$(2,2)$$, $$(3,3)$$. In the ordering on pairs, the interval between $$(2,2)$$ and $$(3,3)$$ is infinite and bounded both above and below. But $$\mathbb{Z}$$ has no infinite interval which is bounded both above and below. So there is no such pairing function.