The Syntax Bridge: A Pragmatic Resolution to the Continuum Hypothesis via Hierarchical Semiotics

The independence of the Continuum Hypothesis (CH) from ZFC set theory stands as one of the most significant epistemological crises in modern mathematics. Traditional interpretations oscillate between Platonist realism—which views independence as a failure to describe a hidden reality—and radical Formalism—which dismisses the question as meaningless. This paper proposes a third path: the “Three-Layer Bridge Model.” We argue that the foundational crisis stems from an unacknowledged Ontological Deficit inherent in any formal description of the infinite. By strictly delineating Context (Authority), Semantics (Structure), and Pragmatics (Use), and redefining Syntax as a functorial bridge (σ ) connecting them, we demonstrate that formal systems must be Anchored to halt the infinite regress of justification. In this framework, the value of CH is determined through an authorized pragmatic assignment based on normative utility, validating techniques like Forcing not as mechanisms of discovery, but as legitimate tools of mathematical construction.