On A Problem Concerning Parameter Free Induction

8 0. Preliminaries In this note we comment on the relationships among local versions of the parameter free induction, collection and pigeonhole schemas. We start by defining the versions that will interest us. We work in the usual language of arithmetic to which a new constant symbol a has been added. P-denotes a finite set of axioms such that if M C P-, then M is the nonnegative part of a commutative discretely ordered ring. Our base theory will be P-+ IZ,, where IZ, denotes the induction schema restricted to Zo formulas with parameters. We recall that in P-+I& every Z,, formula is equivalent to a formula of the form 3XlVx, ... 8, with n alternating quantifiers and Q E 1,.