Looking at some of the comments I realize I might not know what you mean by ordered pair exactly. That said, under one way of interpreting the term "ordered pair" you simply can't usually pick a function at random from the set of all functions from elements of a sequence of the positive integers to the real numbers, such that it can get said to map from an unordered pair of elements of positive integers to a real number. For example, for the subtraction function - with this domain and codomain, it doesn't work out such that -(x, y)=-(y, x). The subtraction function takes ordered pairs such as (3, 4) to a real number. For a binary function B, you need commutation B(x, y)=B(y, x) to hold for the function to meaningfully take an unordered pair to a real number. For a trinary function T you need
T(x, y, z)=T(x, z, y)=T(y, x, z)=T(y, z, x)=T(z, x, y)=T(z, y, x). In general, for an n-ary function you'd need all permutations of variables to come as equivalent, which only holds in exceptional cases.
With N+ denoting the positive integers, you might write f:(N+, N+)->R, such that
for all x, y, f(x, y)=f(y, x). So, the function f takes a member m of N+, and another member n of N+ in that order, such that if m does not equal n, the order in which f took those elements of the domain doesn't matter since f(m, n)=f(n, m). The extra condition (axiom) of commutation or commutativity implies the unorderedness.