Chapter 19: Betweenness from Ultrametric

Tarski’s axiomatization of Euclidean geometry rests on two primitive relations: betweenness and congruence. Classical treatments assume these as undefined primitives. In Category τ, we derive the betweenness relation from the ultrametric distance d(x,y) = 2^{-δ(x,y)} , then prove that it satisfies Tarski’s betweenness axioms T1–T3. Betweenness is a theorem, not an axiom—earned from the CRT tower and the first disagreement depth. The key insight: “y is between x and z” means that x and y agree to greater depth in the primorial tower than x and z do.