Ordinal Folding Index: A Computable Metric for Self-Referential Semantics

TL;DR

The paper introduces the Ordinal Folding Index (OFI), a computable metric that measures the depth of self-reference in statements, linking logic, game theory, and ordinal analysis with practical algorithms and open research questions.

Contribution

It presents OFI as a novel, computable ordinal metric that unifies various theoretical frameworks and provides efficient approximation methods.

Findings

OFI refines classical game-theoretic and logical metrics.

A polynomial-time approximation scheme for OFI on finite arenas.

OFI equals the shortest winning strategy length in evaluation games.

Abstract

The Ordinal Folding Index (OFI) is a new, fully computable yard-stick that measures how many rounds of self-reference a statement, protocol or position must unfold before its truth or outcome stabilises. By turning this abstract 'fold-back' depth into a single ordinal number, OFI forges a direct link between areas that are usually studied in isolation: the closure stages of fixed-point logics, the time-to-win values of infinite parity games, and the ordinal progressions that calibrate the strength of formal theories. We prove that OFI refines all classical game-theoretic and logical metrics while remaining algorithmically enumerable, supply a polynomial-time approximation scheme on finite arenas, and show how the index coincides exactly with the length of the shortest winning strategy in the associated evaluation game. Alongside the theory we outline five open problems from the completeness of the computable-ordinal spectrum to the possibility of 'compressing' deep self-reference that chart a research programme at the intersection of computer-aided logic, algorithmic game theory and ordinal analysis. OFI thus invites game theorists and logicians alike to view infinite play, transfinite induction and reflective reasoning through a single, intuitive lens, opening common ground for techniques.