Chapter 52: The Bounded Powerset

In ZFC, the power set P(X) of a set X is the collection of all subsets of X, and for infinite X it is uncountable by Cantor’s theorem. In τ-set theory, the powerset is defined arithmetically as the set of divisors, and it is always finite. This chapter introduces P_τ(x), the τ-powerset of x ∈ τ-Idx, and proves its finiteness via the divisor function. We show that the Russell paradox cannot arise, prove the well-foundedness of strict membership, and observe a remarkable collapse: the membership relation ∈_τ and the subset relation ⊆_τ coincide.