Formalizing Ordinal Utility

I’ve tried to make an attempt to formalize the Austrian notion of utility. This seems to be a praxeologically sound formalization which does not imply that mathematics can be used in economic theory. This formulation does not state that utility functions are continuous or differentiable, so forming a Walrasian system with them would be nearly impossible. Nevertheless, it is at least an interesting exercise. Feel free to critique it:

The good’s first unit will satisfy the most important end, the second unit will indulge the next most important, the third unit will tend to the third most important, and so on. Thus we get a set of ends E = ( e₁, e₂, e₃, . . . ,e_i) with the following property: For all e₁, e₂, in E, e_j, is preferred to e_i, if and only if i < style="">X = (x₁, x₂, . . . x_i, . . . ) be a set where x, denotes the number of units of good x at a consumer’s disposal. Let x_j > x_i, if and only if j > i. Define a marginal utility function f: X --> R⁺ such that f(x_i) > f(x_j) if and only if x_i, < x_j. This is a result of the fact that a consumer will utilize additional units of good x to satisfy sequentially less important goals. f is thus a decreasing marginal utility function on X . To show that marginal utility is ordinal, let S = (f, g, . . .) be the set of all functions on X defined as above. Then for all f, g in S, and for all x_i, x_j in X, f(x_i) > f(x_j) implies g(x_i) > g(x_j). S is consequently an ordinal measure of marginal utility.