A couple of clarifications if somebody is as confused as me when first reading this.
In ZF we can quantify over sets because “set” is the name we use to designate the underlying objects (the set of natural numbers is an object in the theory). In Peano, the objects are numbers so we can quantify over those we cannot quantify over sets.
Predicates are more “powerful” than first-order formulas so quantifying over predicates allows us to restrict the possible models more than having an axiom for each formula. Even though every predicate is a formula, the interpretation of a predicate is determined by the model so we cannot capture all predicates by having a formula for each predicate symbol.
A couple of clarifications if somebody is as confused as me when first reading this.
In ZF we can quantify over sets because “set” is the name we use to designate the underlying objects (the set of natural numbers is an object in the theory). In Peano, the objects are numbers so we can quantify over those we cannot quantify over sets.
Predicates are more “powerful” than first-order formulas so quantifying over predicates allows us to restrict the possible models more than having an axiom for each formula. Even though every predicate is a formula, the interpretation of a predicate is determined by the model so we cannot capture all predicates by having a formula for each predicate symbol.