Fri Feb 17, 2006
60 Evans Hall, 4:10–6 PM
Toshiyasu Arai (Kobe University)
Resolving the Reflective Universes
Let T denote a set theory for reflecting universes (Richter-Aczel), extending the Kripke-Platek set theory with the Axiom of Infinity. We have designed recursive notation systems Od(T) describing the proof-theoretic ordinals of T.
Our wellfoundedness proof of the notation system Od(T) is based on the maximal distinguished class W (W. Buchholz). It’s a powerful tool for proving wellfoundedness, and is a ??-sub-1-definable set of integers. Thus W is a proper class in T.
To formalize the proof in T, we have to show for each ?? ?? Od(T) there exists a set, say P, such that the maximal distinguished class defined on P has to enjoy the same closure properties as W up to the given ??. Such a set P is said to be ??-Mahlo. Here are involved iterated ??-sub-2-Mahlo operations. It turns out that the ??-Mahlo sets are defined through a ramification process to resolve the reflecting universes for T in terms of iterations of lower Mahlo operations.