Bounded Reach, Unbounded Reflection: A Phenomenological Argument Against the Computability of Human Mathematical Judgment

Benacerraf's opacity objection to the Lucas-Penrose argument holds that even if a Turing machine governs mathematical judgment, we could never identify which one, and so could never construct its Gödel sentence. This paper grants the objection and argues without what it denies. Every $\Pi^1_1$-sound recursively enumerable theory has a proof-theoretic ordinal strictly below the Church-Kleene ordinal $\omega_1^{\mathrm{CK}}$, and this bound follows from the computational hypothesis and a soundness premise alone, whatever the machine may be. Against it stands a datum from the limit stages of reflective ascent: recognising a primitive recursive ordering to be well-founded is the same act at every ordinal height. If first-person experience is veridical concerning the structure of its own acts, and if soundness is given within understanding rather than established from outside, the order types so recognised are cofinal in $\omega_1^{\mathrm{CK}}$, contradicting the bound.