Gödel's Collapse: A Type-Error Interpretation and the Restoration of Hilbert's Program

Gödel’s incompleteness theorems have long been regarded as a decisive barrier to Hilbert’s program, demonstrating that any sufficiently expressive formal system cannot prove its own consistency. This paper challenges that conclusion by identifying a hidden category error in Gödel’s construction. Specifically, the self-referential Gödel sentence G, which asserts its own unprovability, is not a genuine proposition but an ill-typed term—a syntactically malformed expression resulting from an illicit coercion between entities (E) and propositions (P). Within a strictly typed syntactic framework, such as the Pre-Semantic Type Validation (PTV) system, the Gödel sentence cannot even be formed as a well-typed formula. Consequently, both incompleteness theorems collapse: the first loses its object, and the second loses its conditional structure. Consistency thereby becomes a syntactic property by construction, not a semantic hypothesis. The paper concludes that the original aim of Hilbert—to secure the foundations of mathematics through formal means—can be realized within a properly stratified type-theoretic architecture.