The Logical Structure of Physical Laws: A Fixed Point Reconstruction

TL;DR

This work addresses paradoxes in defining physical law via self-reference by replacing naive extensional formulations with a typed fixed-point formalism. Laws are organized as elements of a lattice of theory-packages, and an admissibility operator $\mathcal{F}$—constructed from invariance and symmetry via a Galois connection—yields a fixed-point equation $P=\mathcal{F}(P)$ whose canonical solution is the least fixed point $P=\mu\mathcal{F}$. The framework is instantiated with QED and General Relativity: starting from a minimal seed and applying symmetry, locality (gauge for QED; diffeomorphism invariance for GR), and truncation principles, the least fixed points reproduce the standard QED Lagrangian and the Einstein–Hilbert action with a cosmological term. This approach clarifies how constitutive principles shape the content of physical theories and provides a mathematically controlled way to discuss the closure of theory-packages under admissibility constraints, while noting that the choice of candidate domain and operators remains a substantive physical and philosophical decision.

Abstract

We formalise the self referential definition of physical laws using monotone operators on a lattice of theories, resolving the pathologies of naive set theoretic formulations. By invoking Tarski fixed point theorem, we identify physical theories as least fixed points of admissibility constraints derived from Galois connections. We demonstrate that QED and General Relativity can be represented in such a logical structure with respect to their symmetry and locality principles.

Theorems & Definitions (27)

• Lemma 1
• proof
• Proposition 1
• proof
• Corollary 1
• Definition 1
• Definition 2
• Theorem 1: Tarski's fixed point theorem Tarski1955 in specialised form
• Lemma 2
• proof