Co-Recursive Semantic Self-Extension: A Mathematical Foundation for Open-Ended Intelligence

Abstract

Artificial intelligence has advanced through scale, data, and computation, yet remains bounded by its representational framework. We ask what makes intelligence open-ended. Every inferential system selects meaningful distinctions, limiting what it can learn. Within a fixed purpose, performance reaches a ceiling set by its semantic domain. Teleological succession, in which purposes arise from earlier ones, faces a similar limit: Boolean-closed purpose-transformations exhaust their reachable semantic territory. We shift from purposes to inferential systems. Drawing on Kant and G\"odel, we model cognitive states as partial evaluations over a formal language. Unresolved propositions may be undecided or undecidable. Reasoning can resolve the former but not the latter, leaving a frontier for extension. Non-trivial reductions of undecidable propositions expose undecidable intermediate premises whose reflective reification supports multiple decisive extensions. Finite trace-guided generation remains fallible, requiring meta-level selection and polycontextural exploration. Using Kolmogorov complexity, we show that no architecture with a fixed state description, transition mechanism, and initial condition can generate histories incorporating irreducible algorithmic information. Because physical systems are fixed in this sense, we examine collectives. Coupling finitely many finite agents does not remove the bound. Extension requires persistent external memory preserving semantic content and historical coordinates. Collective generative capacity depends on balancing overlap for transfer with difference for generation. We propose semantic self-extension as a design principle: intelligence should not merely resolve ambiguity within existing meanings, but transform unresolved structures into new spaces of meaning, generating new distinctions, questions, representations, and purposes.