On Wed, 26 Sep 2012 21:21:54 -0700 (PDT), Madhur <firstname.lastname@example.org> wrote in <news:email@example.com> in alt.math.undergrad:
> The natural numbers that we use are said to be derived > from what so called Peano's Axioms.
They *can* be; this is not the only possible formal foundation for them.
> While these axioms (listed below) give a method of > building up counting numbers they do not define or > construct basic arithmetic operations like addition, > subtraction, multiplication, etc or basic comparisons > like that of equality.
Equality is assumed to be a known relation. The arithmetic operations and the linear ordering on the natural numbers are defined using the axioms. This is explained, albeit briefly, in the Wikipedia article on the Peano axioms: