Chapter 50: Membership from Divisibility

Set theory typically begins with an undefined membership relation ∈ governed by axioms. This chapter takes the opposite approach: membership is defined in terms of a relation already earned — internal divisibility . The definition A ∈_τ B ⇔ A ∣ B is simple, but its consequences are far-reaching: membership becomes decidable, the “elements of B” are exactly the divisors of B, and every set-theoretic question reduces to an arithmetic computation on τ-Idx.