Peano Corto and Peano Basso: A Study of Local Induction in the Context ofWeak Theories

摘要

In this paper we study local induction w.r.t. ∑_1-formulas over the weak arithmetic PA~?. The local induction scheme, which was introduced in [7], says roughly this: for any virtual class X that is progressive, i.e., is closed under zero and successor, and for any non-empty virtual class Y that is definable by a ∑_1-formula without parameters, the intersection of X and Y is non-empty. In other words, we have, for all ∑_1-sentences S, that S implies S~X, whenever X is progressive. Since, in the weak context, we have (at least) two definitions of ∑_1, we obtain two minimal theories of local induction w.r.t. ∑_1-formulas, which we call Peano Corto and Peano Basso.