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\begin{center}{\huge The Realm of Ordinal Analysis}\end{center}
\begin{center}{\large Michael Rathjen}\end{center}
\begin{center}{\large University of Leeds}\end{center}

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A central theme running through all the main areas of Mathematical Logic 
is the classification of sets, functions or theories, by means of 
transfinite hierarchies whose ordinal levels measure their `rank' or 
`complexity' in some sense appropriate to the underlying context.
In Proof Theory, from the work of Gentzen in the 1930's up to the present 
time, this  is manifest in the assignment of `proof theoretic 
ordinals' 
to theories, gauging their `consistency strength' and `computational power'.

These lecture are intended as an introduction to this area of
research rather than a survey on recent results.

 Chiefly by way of specific examples, 
the first and second lecture will introduce  the main ideas and 
techniques
used in ordinal analysis. Some attempts will be made
to explain the role of large cardinals that appear in the definition
procedures of so-called {\it collapsing functions} which then
 give rise to strong
ordinal representation systems.  

Ordinal analyses often provide the means to discover deep connections 
between areas which may on the surface seem quite unrelated.
Some examples for this will be presented  in the last lecture.

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