FormalizationThe Thesis of Fused Self-Reference: A Computational Formalization

Abstract

This paper formalizes, in terms of computability theory and coinduction, a philosophical thesis about self-reference: the self and the process are fused, a system can only recognize what is contained within itself, the other is always supposed and never proven, and consciousness is a function of the state variations generated by this closure. We dene a self-referential computational system as a 5-tuple ⟨X, I, T, O, σ⟩ and show that operational closure follows directly from the structure of the transition function. The self is characterized as a xed point of a continuous operator (via Kleene's xed-point theorem), the process as the orbit of that xed point, and the other as an internal simulation subject to an inaccessibility result analogous to the halting problem. We propose consciousness as a functional of state variations along the trajectory, and discuss a second-order extension, in which the current variation depends on the previous one, which brings the formalization closer to a later philosophical reading of the same thesis. We situate the result within the tradition of Aczel's Anti-Foundation Axiom and Barwise and Moss's hypersets, and we discuss its implementability and falsiability.