## Proof-theoretic analysis by iterated reflection

Venue: | Arch. Math. Logic |

Citations: | 3 - 1 self |

### BibTeX

@ARTICLE{Beklemishev_proof-theoreticanalysis,

author = {L. D. Beklemishev},

title = {Proof-theoretic analysis by iterated reflection},

journal = {Arch. Math. Logic},

year = {},

volume = {42},

pages = {2003}

}

### OpenURL

### Abstract

Progressions of iterated reflection principles can be used as a tool for ordinal analysis of formal systems. Technically, in some sense, they replace the use of omega-rule. We compare the information obtained by this kind of analysis with the results obtained by the more usual proof-theoretic techniques. In some cases the techniques of iterated reflection principles allows to obtain sharper results, e.g., to define proof-theoretic ordinals relevant to logical complexity Π 0 1. We provide a more general version of the fine structure formulas for iterated reflection principles (due to U. Schmerl [24]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣn, IΣ − n, IΠ − n and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform Σ1-reflection principle for T is Σ2-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl’s theorem. 1