Bounded minimalisation and bounded counting in argument-bounded idc's

Abstract

We define and investigate a number of small inductively defined classes (idc's),à la Gregorczyk, that are based on argument-bounded initial functions and the bounded minimalisation and bounded counting schemata. We establish equivalences between these and other classes in the literature, with an emphasis on minimalism. We also obtain characterisations of the classes in terms of well-known fragments of first-order predicate logic. † In the expression, the '∃ w6max( x,y) '-quantifier is formal shorthand for the finite disjunction ∃ w6x 1 φ ∨ ∃ w6x k φ ∨ · · · ∨ ∃ w6y φ, which is a PrA-formula when φ is.