trying to make an axiomatic system for "natural" numbers.
natural numbers are the "numbers" that "count" the "finite" "sets". 
0 1 2 3 4 5 6 7 8 9 ...
"zero" is included for the "empty set"
there is no set with 1/2 , -1, sqrt(2) or i members.


(N,+)
a+b=b+a
(a+b)+c=a+(b+c)
a+0=a
if a+b=a+c then b=c
if a+b=0 then a=b=0


(N-{0},.)
ab=ba
(ab)c=a(bc)
a1=a
if ab=ac then b=c
if ab=1 then a=b=1


a(b+c)=ab+ac


a≤b == there exists x in N such that a+x=b
a|b == there exists x in N-{0} such that ax=b


this is a definition: a0=0
the thing that works the same for plus, is ∞ or maybe -∞
doesn't matter