Chapter 72: Proof Theory as 3

Chapters 64–69 built bridges at E₂: they characterised ZFC as a virtual machine and diagnosed incompleteness as the VM’s structural ceiling. This chapter ascends one level. Proof theory does not run proofs—it studies the systems that run proofs. A proof-checker is an E₂ object (it takes a proof string and outputs accept/reject); a proof-theorist is an E₃ object (they model the checker and reason about its limits). Section 1 draws this distinction. Section 2 defines proof theory as E₃ self-modelling applied to E₂ code (the relevant definition, III.D73). Section 3 identifies the diagrammatic sector of E₃—the sector in which Book III itself operates (the relevant definition, III.D74). Section 4 locates the Hinge Theorem as an E₃ statement.