Editing Peano axioms (section)

==== Inequalities ====

The usual [[total order]] relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number:

: For all {{nowrap|''a'', ''b'' ∈ '''N'''}}, {{nowrap|''a'' ≤ ''b''}} if and only if there exists some {{nowrap|''c'' ∈ '''N'''}} such that {{nowrap|1=''a'' + ''c'' = ''b''}}.

This relation is stable under addition and multiplication: for <math> a, b, c \in \N </math>, if {{nowrap|''a'' ≤ ''b''}}, then:

* ''a'' + ''c'' ≤ ''b'' + ''c'', and
* ''a'' · ''c'' ≤ ''b'' · ''c''.

Thus, the structure {{nowrap|('''N''', +, ·, 1, 0, ≤)}} is an [[ordered ring|ordered semiring]]; because there is no natural number between 0 and 1, it is a discrete ordered semiring.

The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤":

: For any [[predicate (mathematics)|predicate]] ''φ'', if
:* ''φ''(0) is true, and
:* for every {{nowrap|''n'' ∈ '''N'''}}, if ''φ''(''k'') is true for every {{nowrap|''k'' ∈ '''N'''}} such that {{nowrap|''k'' ≤ ''n''}}, then ''φ''(''S''(''n'')) is true,
:* then for every {{nowrap|''n'' ∈ '''N'''}}, ''φ''(''n'') is true.

This form of the induction axiom, called ''strong induction'', is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are [[well-order]]ed—every [[empty set|nonempty]] [[subset]] of '''N''' has a [[least element]]—one can reason as follows. Let a nonempty {{nowrap|''X'' ⊆ '''N'''}} be given and assume ''X'' has no least element.

* Because 0 is the least element of '''N''', it must be that {{nowrap|0 ∉ ''X''}}.
* For any {{nowrap|''n'' ∈ '''N'''}}, suppose for every {{nowrap|''k'' ≤ ''n''}}, {{nowrap|''k'' ∉ ''X''}}. Then {{nowrap|''S''(''n'') ∉ ''X''}}, for otherwise it would be the least element of ''X''.

Thus, by the strong induction principle, for every {{nowrap|''n'' ∈ '''N'''}}, {{nowrap|''n'' ∉ ''X''}}. Thus, {{nowrap|1=''X'' ∩ '''N''' = ∅}}, which [[contradiction|contradicts]] ''X'' being a nonempty subset of '''N'''. Thus ''X'' has a least element.