In article <op.wk94sq2i1hq4pq@leon-hp>, wilson <firstname.lastname@example.org> wrote:
> 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 Å~ N Å® 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. > > > sorry. my mistake. Define 2 as SS((0)).
Also, a + 0 = a, instead of a + S(0) = a.
-- --------------------------- | BBB b \ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum videtur. | BBB aa a r bbb | -----------------------------