quality:correction
Equivalent axiomatizations

The above axiomatization of Peano arithmetic uses a signature that only has symbols for zero as well as the successor, addition, and multiplications operations. There are many other different, but equivalent, axiomatizations. One such alternative uses an order relation symbol instead of the successor operation and the language of discretely ordered semirings (axioms 1-7 for semirings, 8-10 on order, 11-13 regarding compatibility, and 14-15 for discreteness):

The theory defined by these axioms is known as PA. It is also incomplete and one of its important properties is that any structure M {\displaystyle M} satisfying this theory has an initial segment (ordered by ≤ {\displaystyle \leq }) isomorphic to N {\displaystyle \mathbb {N} }. Elements in that segment are called standard elements, while other elements are called nonstandard elements.

Finally, Peano arithmetic PA is obtained by adding the first-order induction schema.