On Thu, 27 Sep 2012 01:40:34 -0400, wilson <email@example.com> wrote:
> 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: >> >> <http://en.wikipedia.org/wiki/Peano_axioms?banner=none#Arithmetic> >> >> [...] >> >> Brian > > A bit more complicated: > > First you need another axiom. From the Wikipedia article: > > Addition is the function + : N × N ? N (written in the usual infix > notation, mapping elements of N to other elements of N), defined > recursively as: > > a + S(0) = a > a + S(b) = S(a+b) > Now can define 1 as S(0), 2 as SS()) and 4 as SSSS(0). > > the proof that 2 + 2 = 4 is then a matter of substituting the right > thing in the right place.