Parametrization over inductive relations of a bounded number of variables

We present a Parametrization Theorem for (positive elementary) inductions that use a bounded number of variables. We investigate associated halting problem(s) on classes of finite structures and on solitary ‘unreasonable’ structures. These results involve the complexity of the inductive relations—and the complexity of the structure or class of structures on which these relations live. We also apply this Parametrization Theorem to Moschovakis closure ordinals, to determine when the closure ordinal is greater than ω, and to investigate the closure ordinals of unreasonable structures.