Self-referential theories

TL;DR

This paper explores the structure of self-referential theories in arithmetic, identifying conditions under which such theories are untrue and presenting families that nearly avoid these pitfalls.

Contribution

It introduces a framework for analyzing self-referential theories and characterizes when they are untrue, providing examples of nearly consistent self-referential theories.

Findings

Theories with certain self-referential schemata are untrue.

Some families of self-referential theories can be true if they avoid specific patterns.

The paper delineates the boundary between true and untrue self-referential theories.

Abstract

We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.