Slow Reflection

Abstract

We describe a "slow" version of the hierarchy of uniform reflection principles over Peano Arithmetic ($\mathbf{PA}$). These principles are unprovable in Peano Arithmetic (even when extended by usual reflection principles of lower complexity) and introduce a new provably total function. At the same time the consistency of $\mathbf{PA}$ plus slow reflection is provable in $\mathbf{PA}+\mathrm{Con}(\mathbf{PA})$. We deduce a conjecture of S.-D. Friedman, Rathjen and Weiermann: Transfinite iterations of slow consistency generate a hierarchy of precisely ${\epsilon }_0$ stages between $\mathbf{PA}$ and $\mathbf{PA}+\mathrm{Con}(\mathbf{PA})$ (where $\mathrm{Con}(\mathbf{PA})$ refers to the usual consistency statement).

This version has been accepted for publication in the Annals of Pure and Applied Logic

References in corpus (1)

• Proof Lengths for Instances of the Paris-Harrington Principle

Cited by in corpus (3)

• Proof Lengths for Instances of the Paris-Harrington Principle
• Short Proofs for Slow Consistency
• The Absorption Law, or How to Kreisel a Hilbert-Bernays-Löb