Systemic Constraints of Undecidability

TL;DR

This paper introduces a systemic view of undecidability as a structural property of systems, showing that undecidability propagates through subsystems and constrains prediction and modeling in natural and artificial systems.

Contribution

It develops a theory of systemic undecidability, extending classical computability results into a dynamic systems framework and demonstrating its implications for scientific knowledge boundaries.

Findings

Undecidability is a systemic property, not just a function-specific feature.

Subsystems involved in undecidable systems inherit undecidability.

Architectural innovation cannot circumvent fundamental computational limits.

Abstract

This paper presents a theory of systemic undecidability, reframing incomputability as a structural property of systems rather than a localized feature of specific functions or problems. We define a notion of causal embedding and prove a closure principle: any subsystem that participates functionally in the computation of an undecidable system inherits its undecidability. This result positions undecidability as a pervasive constraint on prediction, modeling, and epistemic access in both natural and artificial systems. Our framework disarms oracle mimicry and challenges the view that computational limits can be circumvented through architectural innovation. By generalizing classical results into a dynamic systems context, this work augments the logical trajectory of G\"odel, Turing, and Chaitin, offering a new perspective of the topology of computability and its interrelation to the boundaries of scientific knowledge.