 Peano arithmetic

Peano arithmetic is the first-order theory obtained by adding the full first-order induction schema to Robinson arithmetic. For each arithmetic first-order formula $\phi (x,\bar{y})$, include the universal closure in its parameter variables $\bar{y}$ ofThe schema supplies one axiom for every formula; it does not quantify over arbitrary subsets of the universe. The order relation can be added by a defining axiom such as $x<y\leftrightarrow \exists z(y=x+Sz)$.

For example, induction on $x$ proves $0+x=x$, complementing the axiom $x+0=x$. Despite its intended interpretation in the natural numbers, Peano arithmetic also has non-standard models of arithmetic by the compactness theorem. First-order induction reaches every definable subset satisfying its hypotheses, without making the theory categorical.^[ref]