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arXiv · 1601.08214

Slow Reflection

Anton Freund
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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 $\mathbf{PA}+\operatorname{Con}(\mathbf{PA})$. 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).

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Anton Freund. 2016-01-29. Slow Reflection. https://doi.org/10.1016/j.apal.2017.06.003

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