Transordinal Fixed-Point Operators and Self-Referential Games: A Categorical Framework for Reflective Semantic Convergence
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Alpay, Faruk Alakkad, Hamdi Al |
| author_facet | Alpay, Faruk Alakkad, Hamdi Al |
| contents | We present a new theoretical framework that unifies category-theoretic fixed-point constructions, transfinite recursion, and game-based semantics to model how interpretations of language can stabilize through unlimited self-reference. By iterating a meaning-refinement operator across all ordinal stages, we isolate a unique "transordinal" fixed point and show, via a hierarchy of reflective games, that this same object is the sole equilibrium of an infinite dialogue between a text and its interpreter. The result delivers a mathematically rigorous account of semantic convergence without resorting to statistical training or empirical benchmarks, yet remains simple to explain: start with a rough meaning, let speaker and listener correct each other forever, and the process provably settles on a single, self-consistent interpretation. Because the construction is entirely symbolic, it offers both precise guarantees for formal linguistics and a blueprint for designing language-aware systems that can reason about their own outputs. The paper details the requisite transordinal machinery, proves existence and uniqueness theorems, and connects them to long-standing questions about reflection, truth, and equilibrium in formal systems and semantics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transordinal Fixed-Point Operators and Self-Referential Games: A Categorical Framework for Reflective Semantic Convergence Alpay, Faruk Alakkad, Hamdi Al Logic in Computer Science 68Q55, 03B70, 47H10, 91A05, 18C15, 68T50 F.3.2; I.2.7; F.1.1; F.4.1 We present a new theoretical framework that unifies category-theoretic fixed-point constructions, transfinite recursion, and game-based semantics to model how interpretations of language can stabilize through unlimited self-reference. By iterating a meaning-refinement operator across all ordinal stages, we isolate a unique "transordinal" fixed point and show, via a hierarchy of reflective games, that this same object is the sole equilibrium of an infinite dialogue between a text and its interpreter. The result delivers a mathematically rigorous account of semantic convergence without resorting to statistical training or empirical benchmarks, yet remains simple to explain: start with a rough meaning, let speaker and listener correct each other forever, and the process provably settles on a single, self-consistent interpretation. Because the construction is entirely symbolic, it offers both precise guarantees for formal linguistics and a blueprint for designing language-aware systems that can reason about their own outputs. The paper details the requisite transordinal machinery, proves existence and uniqueness theorems, and connects them to long-standing questions about reflection, truth, and equilibrium in formal systems and semantics. |
| title | Transordinal Fixed-Point Operators and Self-Referential Games: A Categorical Framework for Reflective Semantic Convergence |
| topic | Logic in Computer Science 68Q55, 03B70, 47H10, 91A05, 18C15, 68T50 F.3.2; I.2.7; F.1.1; F.4.1 |
| url | https://arxiv.org/abs/2507.16620 |