Transfinite Fixed Points in Alpay Algebra as Ordinal Game Equilibria in Dependent Type Theory
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915409451548672 |
|---|---|
| author | Alpay, Faruk Kilictas, Bugra Alpay, Taylan |
| author_facet | Alpay, Faruk Kilictas, Bugra Alpay, Taylan |
| contents | This paper contributes to the Alpay Algebra by demonstrating that the stable outcome of a self referential process, obtained by iterating a transformation through all ordinal stages, is identical to the unique equilibrium of an unbounded revision dialogue between a system and its environment. The analysis initially elucidates how classical fixed point theorems guarantee such convergence in finite settings and subsequently extends the argument to the transfinite domain, relying upon well founded induction and principles of order theoretic continuity.
Furthermore, the resulting transordinal fixed point operator is embedded into dependent type theory, a formalization which permits every step of the transfinite iteration and its limit to be verified within a modern proof assistant. This procedure yields a machine checked proof that the iterative dialogue necessarily stabilizes and that its limit is unique. The result provides a foundation for Alpay's philosophical claim of semantic convergence within the framework of constructive logic. By unifying concepts from fixed point theory, game semantics, ordinal analysis, and type theory, this research establishes a broadly accessible yet formally rigorous foundation for reasoning about infinite self referential systems and offers practical tools for certifying their convergence within computational environments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transfinite Fixed Points in Alpay Algebra as Ordinal Game Equilibria in Dependent Type Theory Alpay, Faruk Kilictas, Bugra Alpay, Taylan Logic in Computer Science Artificial Intelligence 68T27, 03B70, 68Q55 This paper contributes to the Alpay Algebra by demonstrating that the stable outcome of a self referential process, obtained by iterating a transformation through all ordinal stages, is identical to the unique equilibrium of an unbounded revision dialogue between a system and its environment. The analysis initially elucidates how classical fixed point theorems guarantee such convergence in finite settings and subsequently extends the argument to the transfinite domain, relying upon well founded induction and principles of order theoretic continuity. Furthermore, the resulting transordinal fixed point operator is embedded into dependent type theory, a formalization which permits every step of the transfinite iteration and its limit to be verified within a modern proof assistant. This procedure yields a machine checked proof that the iterative dialogue necessarily stabilizes and that its limit is unique. The result provides a foundation for Alpay's philosophical claim of semantic convergence within the framework of constructive logic. By unifying concepts from fixed point theory, game semantics, ordinal analysis, and type theory, this research establishes a broadly accessible yet formally rigorous foundation for reasoning about infinite self referential systems and offers practical tools for certifying their convergence within computational environments. |
| title | Transfinite Fixed Points in Alpay Algebra as Ordinal Game Equilibria in Dependent Type Theory |
| topic | Logic in Computer Science Artificial Intelligence 68T27, 03B70, 68Q55 |
| url | https://arxiv.org/abs/2507.19245 |