On Thu, 27 Sep 2012 01:40:34 -0400, wilson <email@example.com> wrote in <news:op.wk94pwvp1hq4pq@leon-hp> in alt.math.undergrad:
> On Thu, 27 Sep 2012 01:18:28 -0400, Brian M. Scott <firstname.lastname@example.org> > wrote:
>> On Wed, 26 Sep 2012 21:21:54 -0700 (PDT), Madhur >> <email@example.com> wrote in >> <news:firstname.lastname@example.org> >> 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: