Chapter 19: Worlds, Topos, and Truth-Makers

The topos structure internal to τ provides a natural setting for truth, possibility, and truth-making. The subobject classifier Ω furnishes a richer space of truth values than the classical two-element set {0, 1}. Possible worlds are reconstructed as internal domains—objects of the presheaf topos with accessibility morphisms between them—without invoking Lewisian realism about concrete possible worlds. Truth-makers are classified into four structural types: inclusions, sections, commuting diagrams, and invariants. The Coherence-Correspondence Unification Theorem (VII.T11) shows that the sheaf condition on a Grothendieck site simultaneously encodes local-to-global coherence and section-existence correspondence, thereby unifying two historically rival theories of truth within a single structural framework.