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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.02613 |
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Table of Contents:
- We construct a rigorous mathematical framework for an abstract continuous evolution of an internal state, inspired by the intuitive notion of a flowing thought sequence. Using tools from topology, functional analysis, measure theory, and logic, we formalize an indefinitely proceeding sequence of states as a non-linear continuum with rich structure. In our development, a trajectory of states is modeled as a continuous mapping on a topological state space (a potentially infinite-dimensional Banach space) and is further conceptualized intuitionistically as a choice sequence not fixed in advance. We establish fundamental properties of these state flows, including existence and uniqueness of evolutions under certain continuity conditions (via a Banach fixed-point argument), non-measurability results (demonstrating the impossibility of assigning a classical measure to all subsets of the continuum of states), and logical semantic frameworks (defining a Tarskian truth definition for propositions about states). Throughout, we draw on ideas of Brouwer, Banach, Tarski, Poincaré, Hadamard, and others--blending intuitionistic perspective with classical analysis--to rigorously capture a continuous, ever-evolving process of an abstract cognitive state without resorting to category-theoretic notions.