 results in (first order) Peano Arithmetic (PA). Note that unlike Q, PA contains infinitely many axioms, because all (infinitely many) instances of the induction scheme, one corresponding to every formula \(\phi(x)\) (with at least one free variable) of the language, are taken as axioms. But it is a routine mechanical task to check whether a given sentence is an instance of this scheme. PA is generally taken as the standard first-order system of arithmetic.