[Phys-L] Re: arithmetic

["John S. Denker" <jsd@AV8N.COM>]

On 02/23/05 15:07, RAUBER, JOEL wrote:
  > Should axiom (4) read
> (a * Sb) = ((a * b) + a)

Yes.  Sorry.  That's why I like to say everything twice.
The reference I gave gets it right:
   http://www.answers.com/topic/peano-axioms

> Also, should commutivity of the operations "*" and "+" be added as an
> axiom?

That's more interesting.  As a practical matter, there's
no harm in doing so ... but if we put on our formal
abstract elegant mathematician hats, we will choose not
to do so.  It's against the spirit of mathematics to
make a new axiom of something that could have been
derived from the old axioms.

The proof that (S0 + 0) is equal to (0 + S0) is easy.

By way of analogy, you may recall that on Day One of
the group-theory class they stipulated the existence
of "some" right identity and "some" right inverse for
each element.  Then for homework you had to prove that
the identity was unique and the same element worked
as a left identity, and for each element its inverse
was unique and worked as a left inverse.  They "could"
have stipulated the existence of a unique left-and-right
identity, but they didn't.

   BTW I think that the axiomatic approach is somewhat
   overrated as a model of knowledge and thinking in
   general ... it is a bit of an artificial game.  But
   it is sometimes an amusing game, and if we are going
   to play it we ought to play by its rules and traditions.