# A note on the consistency operator

## TL;DR

This paper discusses the role of the consistency operator in formal theories, showing it as the minimal natural extension method that preserves the ordering by consistency strength.

## Contribution

It formalizes and proves that the consistency operator is the weakest natural method for uniformly extending axiomatic theories.

## Key findings

- Consistency operator is the weakest natural extension method.
- Natural theories are pre-well-ordered by consistency strength.
- Adding the consistency statement yields the next strongest theory.

## Abstract

It is a well known empirical observation that natural axiomatic theories are pre-well-ordered by consistency strength. For any natural theory $T$, the next strongest natural theory is $T+\mathsf{Con}_T$. We formulate and prove a statement to the effect that the consistency operator is the weakest natural way to uniformly extend axiomatic theories.