Chaos Without Infinity: How Computational Finitism Resolves the Paradox of Deterministic Chaos

Abstract

Deterministic chaos has long been considered a hallmark of continuous nonlinear systems, characterized by sensitive dependence on initial conditions and positive Lyapunov exponents implying infinite divergence. We challenge this paradigm through the lens of Computational Finitism, demonstrating that when chaotic systems are implemented on a finite-alphabet substrate (digits 0-9, or more generally 0 to B-1), true mathematical chaos becomes impossible. Through five comprehensive simulations, Cellular Automaton predictability horizons, Logistic Map bifurcation truncation, Hénon Map quantization, Lorenz Attractor discretization, and Mandelbrot Set boundary truncation, we prove that finite state spaces force all dynamics toward structural completion (fixed points or limit cycles) rather than infinite divergence. The apparent chaos observed in physical systems is not true mathematical chaos but computational irreducibility within an astronomically large but strictly bounded state space. We introduce the concept of the Finitism Predictability Horizon, where exponential divergence saturates at the hardware resolution limit, and demonstrate that "infinite fractal detail" is a representational artifact requiring infinite precision. These results establish that chaos theory must be reformulated within finite computational constraints, resolving the tension between deterministic laws and unpredictable behavior without invoking actual infinities. Finitism is not a hammer that sees nails everywhere, but it is the answer to a world that assumed that everything is infinite, when it is not, and will not be.