\documentstyle{article}

\title{Introducing the hardline}
\author{Toshiyasu Arai\\
Faculty of Integrated Arts and Sciences\\
Hiroshima University\\
Higashi-Hiroshima, 739 Japan\\
{\it E-mail}:arai@mis.hiroshima-u.ac.jp}
\date{}
\begin{document}
\maketitle

G. Gentzen \cite{G3} published his new version of consistency proof for
first order number theory in 1938. He had already had two consistency
proofs \cite{G1} and \cite{G2}. The first used {\em constructive but rather
abstract notion of functionals}. In the second he had first introduced {\em
transfinite ordinals} in proof theory. Although he formulated the result as
a consistency proof, his interest seems to involve a taking off from
Hilbert's program. As to this turning G. Kreisel \cite{K} p. 262 wrote: 

\begin{quotation}
$\ldots$, by introducing a {\em quantitative ordinal measure} he (=Gentzen)
forces us to pay attention to combinatorial complexity and thereby makes it
at least more difficult for us to slip into an abstract reading.
\end{quotation}

It seems that the purpose of the third "Neue Fassung" is to make a lucid
exposure of this {\em combinatorial complexity} which Gentzen discovered in
{\em finite proof figures} of number theory.
G. Takeuti \cite{T2} followed this idea and developed a proof theory of
systems of second order arithmetic including $\Pi^1_1$-Comprehension Axiom,
$\Pi^1_1-CA$.
We follow in the wake of Gentzen and Takeuti.

In this talk we survey the results in proof theory for theories of
recursively large ordinals.

\begin{thebibliography}{99}
\bibitem{G1} G. Gentzen, Der erste Widerspruchsfreiheitsbeweis f\"ur die
klassische Zahlentheorie, Archiv f\"ur mathematische Logik und
Grundlagenforschung, vol. 16 (1974), 97-118.
\bibitem{G2} G. Gentzen, Die Widerspruchsfreiheit der reinen Zahlentheorie,
Mathematische Annalen, vol. 112 (1936), 493-565.
\bibitem{G3}  G. Gentzen, Neue Fassung des Widerspruchsfreiheitsbeweises
f\"ur die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung
der exakter Wissenschaften, Neue Folge, vol. 4 (1938), 19-44.
\bibitem{K} G. Kreisel, Review of the book 'The Collected Papers of Gerhard
Gentzen', ed. and transl. by M. E. Szabo, Journal of Philosophy, vol. 68
(1971), 238-265.
\bibitem{T2} G. Takeuti, Proof Theory, second edition (North-Holland,
Amsterdam, 1987).
\end{thebibliography}

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