Using Padoa’s principle to prove the non-definability, in terms of each other, of the three fundamental qualitative concepts of comparative probability, independence and comparative uncertainty, with some new axioms of qualitative independence and uncerta

• Mutual non-definability of the three fundamental qualitative probability concepts.
• Representation theorem for comparative probability as an extensive quantity.
• Random-variable axioms of independence in terms of indicator functions.
• Qualitative uncertainty as a fundamental probability concept measured by entropy.

First, Padoa’s principle is used to prove the non-definability of the fundamental qualitative concepts of comparative probability, independence and comparative uncertainty in terms of each other. Second, the qualitative axioms of uncertainty leading to an entropy representation are new. Third, a qualitative random-variable axiomatization of these concepts is given, but the random variables are restricted to generalized indicator functions, their products and their iterates. A new axiom of independence in terms of such indicator functions is used in this axiomatization. Fourth, a standard extensive-quantity representation is then proved for comparative probability, and the new axiom of independence provides the basis for proving the desired absolute invariance theorem for the constructed probability measure.