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A. Beckmann:
"Dynamic ordinal analysis";
Archive for Mathematical Logic, 42 (2003), 303 - 334.

Dynamic ordinal analysis is ordinal analysis for weak
arithmetics like fragments of bounded arithmetic. In this paper
we will define dynamic ordinals -- they will be sets of number
theoretic functions measuring the amount of $s\Pi^b_1$ order
induction available in a theory. We will compare order induction
to successor induction over weak theories. We will compute
dynamic ordinals of the bounded arithmetic theories
$s\Sigma^b_n(X)L^mIND$ for $m=n$ and $m=n+1$, $n\ge0$. Different
dynamic ordinals lead to separation. Therefore, we will obtain
several separation results between these relativized theories. We
will generalize our results to arbitrary languages extending the
language of Peano arithmetic.