On the Decidability of Bounded Arithmetic Hilbert's Entscheidungsproblem Revisited

We examine Hilbert’s Entscheidungsproblem within bounded arithmetic, using the algebraic characterization of polynomial functions as the smallest class closed under addition and multiplication from generators {0, 1, n}. We show that every predicate definable by a bounded formula is determinate: its truth value for any input is fixed by arithmetic and requires no unbounded process to ascertain. The undecidability results of G¨odel, Turing, and Church are identified as consequences of unbounded quantification—structure imported into arithmetic, not supplied by it. Within the algebraic spine, Hilbert’s program succeeds.