\documentstyle{article}
\begin{document}
\title{How is it that infinitary methods can be applied
to finitary mathematics?}
\author{Andreas Weiermann\\
Institut f\"ur mathematische Logik und Grundlagenforschung\\
der Westf\"alischen Wilhelms-Universit\"at M\"unster\\
Einsteinstr. 62,
D-48149 M\"unster,
Germany,\\
{\tt email: weierma@math.uni-muenster.de}\\
{\tt URL: http://wwwmath.uni-muenster.de/math/inst/logik/}\\
{\tt org/staff/weiermann/index.html}}
\date{}
\maketitle
We outline a vision of
how to apply a certain
type of infinitary methods to questions
of finitary mathematics 
and
we are going to survey the following topics.\\
{\bf 1. Applications of infinitary proof theory to the 
proof theory of first order Peano arithmetic.}
We describe how Pohlers's method of local predicativity
and Buchholz' method of operator controlled derivations
yield genuine characterizations of the provably
recursive functions of $PA$.\\
{\bf 2. Applications of methods from infinitary proof theory to primitive
recursion (in higher types).}
Using Sch\"utte's 1977 exposition
of Howard's 1970 proof
in combination with an appropriate Mostowski collapse operation
we are going to define by constructive means (of a lowest
possible proof-theoretic complexity) a function
$I$ from the terms of G\"odel's $T$ into the 
set of natural numbers such that if a term $a$ 
reduces to a term $b$ then $I(a)>I(b)$.
This recent result (to appear in JSL) 
strengthens considerably -- in a condensed
and compact form -- a variety of known results
on questions of (strong) normalization for G\"odel's $T$.\\
{\bf 3. Applications of concepts from infinitary
proof theory to
subrecursion theory and the rewrite system for
the Ackermann function.}
We show that there exist at least three
non-pathological but genuinely different
slow growing hierarchies $(G_\alpha)_{\alpha<\varepsilon_0}$.
These investigations indicate that the notion
of natural assignment of fundamental sequences 
for the limits less than $\varepsilon_0$
is deeply connected to Lipshitz-continuity 
properties
of the binary ordinal addition function.
If time is left we shall discuss the merits
of a 1977 paper by Vogel
on the slow growing hierarchy
and we discuss some interesting applications of Vogel's
paper to the rewrite system
for the Ackermann function.
\end{document}