Quantifiers in Limits

The standard definition of limz→∞ F (z) = ∞ is an ∀∃∀ sentence. Mostowski showed that in the standard model of arithmetic, these quantifiers cannot be eliminated. But Abraham Robinson showed that in the nonstandard setting, this limit property for a standard function F is equivalent to the one quantifier statement that F (z) is infinite for all infinite z. In general, the number of quantifier blocks needed to define the limit depends on the underlying structure M in which one is working. Given a structure M with an ordering, we add a new function symbol F to the vocabulary of M and ask for the minimum number of quantifier blocks needed to define the class of structures (M, F ) in which limz→∞ F (z) = ∞ holds. We show that the limit cannot be defined with fewer than three quantifier blocks when the underlying structureM is either countable, special, or an o-minimal expansion of the real ordered field. But there are structures M which are so powerful that the limit property for arbitrary functions can be defined in both two-quantifier forms.