Axiomatic Origins of Mathematical Entropy: Grading Ordered Sets

Shannon's entropy and other entropy-based concepts are derived from the new, more general concept of relative divergence of one "grading' function on a linearly ordered set from another such function. The definition of relative divergence is derived based on "common sense' assumptions about comparing grading functions. Shannon's entropy formulas emerge from the respective relative divergence ones, entropy based methods are extended to more general cases and some new applications.