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. -- Using Opera's revolutionary e-mail client: http://www.opera.com/mail/