The Inaccessible Game

The No-Barber Principle

Russell’s Barber Paradox:

Barber shaves all who don’t shave themselves

Does the barber shave themselves?

Paradox: Definition includes itself in scope

The Munchkin Provision

Munchkin Card Game (Jackson-munchkin01?):

Rules may be inconsistent

Resolution: “Loud arguments, with owner having last word”

For foundations: Need something better!

No external referee for mathematics

No External Adjudicators

No-Barber Principle:

Rules must be internally adjudicable

Forbidden: * External observer * Pre-specified outcome space * Privileged decomposition * External time parameter

No appeal to structure outside the game

Entropic Exchangeability

Entropic Exchangeability:

Admissible rules must: 1. Use only reduced descriptions 2. Be relabeling-invariant 3. Not require global distinguishability

What This Excludes

Violations of No-Barber: * Partial conservation (some variables isolated) → privileges variables * Time-varying \(C\) → needs external clock * Observer-relative isolation → needs external observer * Probabilistic isolation → needs external measure

All smuggle in external structure

Foundations: Information Loss and Entropy

The Inaccessible Game Setup

{Inspired by the no-barber principle, we set up the game in a way that attempts to avoid “external structure”. The first two things we need to do this are 1. A representation of information loss 2. A prohibition of information exchange with the game

At this point there’s a challenge, how do we obtain a representation of information loss without including external structure? Our best suggestion is the axiomatic frameworks of Baez et al (Baez et al. (2011)) and Parzygnat ((Parzygnat?)

Open Questions

Open Questions:

• Formalise no barber principle
• What is the stage/game board/space
• Much to explore