A Remark on Independence Results for Sharply Bounded Arithmetic

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]. By sharply bounded arithmetic we understand fragments of the theory S 1 2 of Bounded Arithmetic of Buss 1] given by the quantiier-free BASIC axioms and some induction scheme for sharply bounded formulas, i.e. for formulas containing only quantiiers of the form 8xjtj and 9xjtj. The class of sharply bounded formulas is denoted by b 0. In particular, S 0 2 is the theory deened by BASIC and the scheme b 0-P IND