The systems Kα of transfinite cumulative types up to α are extended to systems K∞α that include a natural infinitary inference rule, the so-called limit rule. For countable α a semantic completeness theorem for K∞α is proved by the method of reduction trees, and it is shown that every model of K∞α is equivalent to a cumulative hierarchy of sets. This is used to show that several axiomatic first-order set theories can be interpreted in K∞α, for suitable α.

We define a property of substructures of models of arithmetic, that of being length-initial, and show that sharply bounded formulae are absolute between a model and its length-initial submodels. We use this to prove independence results for some weak fragments of bounded arithmetic by constructing appropriate models as length-initial submodels of some given model.

We prove some independence results for the bounded arithmetic theoryR 2 0 , and we define a class of functions that is shown to be an upper bound for the class of functions definable by a certain restricted class of ∑ 1 b in extensions ofR 2 0.

The theories Si1 and Ti1 are the analogues of Buss' relativized bounded arithmetic theories in the language where every term is bounded by a polynomial, and thus all definable functions grow linearly in length. For every i, a Σbi+1-formula TOPi, which expresses a form of the total ordering principle, is exhibited that is provable in Si+11 , but unprovable in Ti1. This is in contrast with the classical situation, where Si+12 is conservative over Ti2 w. r. t. Σbi+1-sentences. The independence (...) results are proved by translations into propositional logic, and using lower bounds for corresponding propositional proof systems. (shrink)

The purpose of this note is to show that the independence results for sharply bounded arithmetic of Takeuti [4] and Tada and Tatsuta [3] can be obtained and, in case of the latter, improved by the model-theoretic method developed by the author in [2].