When is arithmetic possible? (Q922533)

When a structure or class of structures admits an unbounded induction, arithmetic can be done on the stages of that induction; if only bounded inductions are admitted, then every inductively definable relation can be defined by a finite explicit expression. This article presents evidence that the converse is true, and investigates a combinatorial property equivalent to ``all \(L^{<\omega}_{\infty \omega}\)-definable relations are elementary''.