The Realm of Ordinal Analysis

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Reflection principles and provability algebras in formal arithmetic (2005, background): “For many concrete theories these ordinals can be explicitly calculated, which constitutes the subject of the so-called ordinal analysis of axiomatic systems [67, 72, 73].”
The art of ordinal analysis (2006, background): “A detailed account of these results has been given in [31], section 3.”