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 $\mathbf{ PA}+\operatorname{ Con}(\mathbf{ PA})$. 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 $\mathbf{ PA}+\operatorname{ Con}(\mathbf{ PA})$ (where $\operatorname{ Con}(\mathbf{ PA})$ refers to the usual consistency statement).
Categories: math.LO