A new starting axiomatic for numbers

Hi,
This post is about suggesting a (maybe new ?) kind of maths with a different start for integers to your appreciation (that's why I put it speculations)

The idea is to modify Peano, to create a set, with a start and a end, (no, it's not really modular arithmetic with unknown modulus)

So we create two extremum :

• 0 at left, inf at right
• the function succ(x) 
  • exist y=succ(x) for x != inf
• the function prec(x)
  • exist y=prec(x) for x != 0

We have addition etc, subtraction multiplication etc. all within the limit for "succ" that imply that operations are have no solutions close to inf

every relation between elements have a "anti" counterpart, using the function y= inf -x
so we have anti multiplication, anti addition etc:

• x '* y = inf - ((inf-x) * (inf - y)) 
• x '+ y = inf - ((inf-x) + (inf - y)) =  x+ y - inf
• 0 is a multiple of any x (and anti-multiple)
• inf is a multiple of any x (and anti-multiple)

etc etc

And then we can extend numbers on both side : 

• negative numbers as solutions for
  0 - x
  on the left
• some other numbers as solutions for 
  inf + x 
  on the right

we have "antiprimes" ( { inf - primes } ) ... antifractions etc

such a symmetrical axiomatic has to be useful in some ways.. ?