When is arithmetic possible?

Abstract

When a structure or class of structures admits an unbounded induction, we can do arithmetic on the stages of that induction: if only bounded inductions are admitted, then clearly each inductively definable relation can be defined using a finite explicit expression. Is the converse true? We examine evidence that the converse is true, in positive elementary induction (where explicit = elementary). We present a stronger conjecture involving the language L consisting of all L ∞ω formulas with a finite number of variables, and examine a combinatorial property equivalent to “all L-definable relations are elementary”.