The Abstract Mathematics of the Axiomatic Method

We study the axiomatic method itself as a mathematical object. An axiom is a subset of a finite domain, an axiom system is a family of such subsets, and the model class is their intersection. The model-class map forms a Galois connection with the theory map, and adding independent axioms drives a strictly decreasing chain of model classes terminating in one of four regimes: vacuity, possibility, uniqueness, or impossibility. Given a probability measure on the domain, every axiom system defines a cooperative game on its axiom set, and the Shapley value of this game gives a canonical attribution of credit or blame to each axiom. We prove that axiom games are characterized by nonnegative Möbius coefficients of the complement function 1-v, that all pairwise axiom interactions exhibit diminishing marginal returns, and that the minimum number of axiom drops needed to restore possibility equals the transversal number of the family of impossibility cores. We introduce the impossibility dimension, prove a duality relating it to the minimum domain size certifying independence, show that the simplicial complex of possible sub-systems has homology governed by the Alexander dual of the core family, and establish a sharp phase transition from possibility to impossibility in random axiom systems. The framework is illustrated on a catalogue of fully derived examples, including Condorcet, Arrow-type, and Nash-type systems on small finite domains, in which every model-class size is obtained by direct enumeration.