Dynamic ordinal analysis

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 sΠb1(X) 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 sΣbn(X)-L IND for m = n and m = n+1, n ≥ 0. Different dynamic ordinals lead to separation. In this way we will obtain several separation results between these relativized theories. We will generalize our results to further languages extending the language of bounded arithmetic.