The Model Theory of Artificial Intelligence: Jonsson Semantic Spaces and Inference as Type Realization

Despite the empirical success of Large Language Models (LLMs) in demonstrating emergent cognitive abilities, a rigorous mathematical foundation explaining their internal inference mechanisms remains elusive. In this paper, we propose a novel foundational framework that maps the autoregressive generation process of AI to the realization of logical types within the context of Jonsson theories. We formalize the latent semantic space of an LLM as an $\omega^+$-homogeneous-universal semantic model $C_{\mathcal{T}}$ of an underlying inductive theory $\mathcal{T}$ acquired during pre-training. We hypothesize that the realization of Artificial General Intelligence (AGI) is mathematically equivalent to achieving a saturated semantic model of the theory's center.