Chapter 57: Approaches to Infinity

the relevant chapter showed that Cantor’s diagonal argument cannot execute in τ: the bounded powerset, the absence of impredicative comprehension, and the diagonal discipline together block the mechanism that produces uncountable sets. But blocking the diagonal is a negative result — it says what τ does not have. This chapter provides the positive replacement. The “structure of infinity” that analysis needs comes not from different sizes of infinity (the cardinality hierarchy ℵ_0 < ℵ_1 < ℵ_2 < ⋯) but from different approaches to the unique infinity ω. The omega-germs of the relevant chapter are precisely these approaches: each germ is a direction of convergence toward the beacon, and the collection of all approaches is the boundary ring ℤ_τ . Cardinality is unnecessary; convergence suffices.