Editing Peano axioms (section)

=== 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{{sfn|Kaye|1991|pages=16–18}} uses an order relation symbol instead of the successor operation and the language of [[Semiring#Discretely ordered semirings|discretely ordered semirings]] (axioms 1-7 for semirings, 8-10 on order, 11-13 regarding compatibility, and 14-15 for discreteness):
<!-- These axioms are taken directly from Kaye 1991. Please don't "tweak" them, add additional axioms, or remove axioms without discussion on the talk page. -->
# <math>\forall x, y, z \ ( (x + y) + z = x + (y + z) )</math>, i.e., addition is [[associative property|associative]].
# <math>\forall x, y \ ( x + y = y + x )</math>, i.e., addition is [[commutative property|commutative]].
# <math>\forall x, y, z \ ( (x \cdot y) \cdot z = x \cdot (y \cdot z) )</math>, i.e., multiplication is associative.
# <math>\forall x, y \ ( x \cdot y = y \cdot x )</math>, i.e., multiplication is commutative.
# <math>\forall x, y, z \ ( x \cdot (y + z) = (x \cdot y) + (x \cdot z) )</math>, i.e., multiplication [[distributive property|distributes]] over addition.
# <math>\forall x \ ( x + 0 = x \land x \cdot 0 = 0 )</math>, i.e., zero is an [[identity element|identity]] for addition, and an [[absorbing element]] for multiplication (actually superfluous{{NoteTag|"<math> \forall x \ ( x \cdot 0 = 0 ) </math>" can be proven from the other axioms (in first-order logic) as follows. Firstly, <math> x \cdot 0 + x \cdot 0 = x \cdot (0+0) = x \cdot 0 = x \cdot 0 + 0 </math> by distributivity and additive identity. Secondly, <math> x \cdot 0 = 0 \lor x \cdot 0 > 0 </math> by Axiom 15. If <math> x \cdot 0 > 0 </math> then <math> x \cdot 0 + x \cdot 0 > x \cdot 0 + 0 </math> by addition of the same element and commutativity, and hence <math> x \cdot 0 + 0 > x \cdot 0 + 0 </math> by substitution, contradicting irreflexivity. Therefore it must be that <math> x \cdot 0 = 0 </math>.}}).
# <math>\forall x \ ( x \cdot 1 = x )</math>, i.e., one is an [[identity element|identity]] for multiplication.
# <math>\forall x, y, z \ ( x < y \land y < z \Rightarrow x < z )</math>, i.e., the '&lt;' operator is [[Transitive relation|transitive]].
# <math>\forall x \ ( \neg (x < x) )</math>, i.e., the '&lt;' operator is [[Reflexive relation|irreflexive]].
# <math>\forall x, y \ ( x < y \lor x = y \lor y < x )</math>, i.e., the ordering satisfies [[trichotomy (mathematics)|trichotomy]].
# <math>\forall x, y, z \ ( x < y \Rightarrow x + z < y + z )</math>, i.e. the ordering is preserved under addition of the same element.
# <math>\forall x, y, z \ ( 0 < z \land x < y \Rightarrow x \cdot z < y \cdot z )</math>, i.e. the ordering is preserved under multiplication by the same positive element.
# <math>\forall x, y \ ( x < y \Rightarrow \exists z \ ( x + z = y ) )</math>, i.e. given any two distinct elements, the larger is the smaller plus another element.
# <math>0 < 1 \land \forall x \ ( x > 0 \Rightarrow x \ge 1 )</math>, i.e. zero and one are distinct and there is no element between them. In other words, 0 is [[Covering relation|covered]] by 1, which suggests that these numbers are discrete.
# <math>\forall x \ ( x \ge 0 )</math>, i.e. zero is the minimum element.

The theory defined by these axioms is known as '''PA<sup>−</sup>'''. It is also incomplete and one of its important properties is that any structure <math>M</math> satisfying this theory has an initial segment (ordered by <math>\le</math>) isomorphic to <math>\N</math>. 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.