The No Barber Principle: Towards Formalised Selection in the Inaccessible Game

TL;DR

This paper introduces the no-barber principle, a formal rule for the inaccessible game that prevents external structures and impredicative definitions, aiming to identify suitable internal dynamics within complex information-theoretic systems.

Contribution

It formulates the no-barber principle to restrict dynamics that rely on external structures, linking it to categorical properties and entropy constraints in the inaccessible game.

Findings

Classical category FinProb is incompatible with the no-barber principle due to its cartesian structure.

Noncommutative category NCFinProb aligns with the no-barber principle, lacking canonical copying maps.

The no-barber principle prevents impredicative definitions and external adjudicators in the system.

Abstract

The inaccessible game (Lawrence, 2025, 2026) is an information-theoretic dynamical system governed by three information loss axioms, a marginal entropy conservation constraint and maximum entropy dynamics. In this paper we look at selection in the game. Our aim is to develop a selection policy for the game rules based on a minimal set of assumptions. We seek necessary consistency constraints for self-determining dynamical systems. Specifically, we suggest that rules that quantify over distinctions they cannot internally represent risk impredicative-style circularity. Our criterion is motivated by an analogy with Russell's paradox. We formulate a no-barber principle which prohibits dynamics that appeal to external adjudicators or structure lying outside the system. To motivate our principle we examine Russell's paradox through its structural formalisation as a Lawvere diagonalisation. The marginal-entropy conservation in the game is a nontrivial entropy constraint which prohibits external structure. Through the no-barber principle we argue (i) the classical category FinProb, in which Shannon entropy is characterised, is cartesian and provides canonical diagonal (copying) maps that make Lawvere-style constructions expressible and is structurally incompatible with the no-copying instantiation of the no-barber principle studied here. (ii) the noncommutative category NCFinProb, in which von Neumann entropy is characterised, is symmetric monoidal and lacks canonical copying maps, making it a more natural candidate for the game's internal language.