A new perspective on completeness and finitist consistency.

This article investigates metamathematical properties of consistent arithmetical theories \( T \) containing the fragment \(\textsf{I}\Sigma_1\), focusing on proof predicates derived from various numerations of axioms. The main results establish (1) **Numeral Completeness**: for every true arithmetical sentence with a \(\Sigma_1(\textsf{I}\Sigma_1)\)-formula, there exists a numeration \(\tau\) of \(T\)'s axioms such that \(\textsf{I}\Sigma_1\) proves the provability of the formula’s numeral instances under \(\tau\); (2) **Numeral Consistency**: for any consistent theory \(T\) with a \(\Delta_1(\textsf{I}\Sigma_1)\)-proof predicate, there is a \(\Sigma_1(\textsf{I}\Sigma_1)\)-numeration \(\iota\) of \(\textsf{I}\Sigma_1\) such that \(\textsf{I}\Sigma_1\) proves that no numeral codes a proof of contradiction according to that predicate; and (3) **Partial Finitism**: there exists a primitive recursive function producing, for each natural number \(n\), an \(\textsf{I}\Sigma_1\)-proof that the \(n\)-th numeral is not a proof of contradiction. These results complement Gödel’s Incompleteness Theorems by showing that while universal completeness and consistency proofs are impossible, existential versions—guaranteeing the existence of suitable codifications and numerations—are achievable within a finitist framework. The work connects these findings to Hilbert’s consistency program (Hilbert’s program) and his reflections after 1931, illustrating a nuanced fulfillment of Hilbert’s goals through numeral-wise provability and consistency statements in modest arithmetic theories.