Slow Reflection

Anton Freund

Published 2016-01-29Version 1

We describe a "slow" version of the hierarchy of uniform reflection principles over Peano Arithmetic. 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 $\mathbf{PA}+\text{Con}(\mathbf{PA})$. We deduce a conjecture of S.-D. Friedman, Rathjen and Weiermann: Transfinite iterations of slow consistency generate a hierarchy of precisely $\varepsilon_0$ stages between $\mathbf{PA}$ and $\mathbf{PA}+\text{Con}(\mathbf{PA})$ (where $\text{Con}(\mathbf{PA})$ refers to the usual consistency statement).