Validity of ‘generic n’ arguments near the largest expressible natural number

Ascertain whether the standard diagonalization-style reasoning that rules out an n-bit program deciding halting for all n-bit programs should be accepted for generic n when n approaches the largest expressible natural number under finite information constraints.

Background

The author considers a finite-information viewpoint in which there may be a largest expressible natural number. Within this framework, classical diagonalization arguments used in theorems like the Space Hierarchy Theorem suggest that, for a fixed size bound n, no program can decide halting for all programs of size n.

While such arguments hold for modest n, the author notes that when n nears the largest expressible number, the argument cannot literally be carried out, raising doubt about whether the ‘generic n’ reasoning remains justified in that regime.