Should I start another elementary mathematical topic (and which) or should I stop writing, and spend some time now on learning CSS (Cascading Style Sheets), and then JavaScript (in this order, because they are presented in my textbook in this order, and I’d like to be systematic, at least on this occasion)?

One of the topics is the beginning Number Theory. It’s rather clear that One should not start with Peano axioms, followed by the inductive definitions of the operations. It’d take too long and be too boring to get to the first juicy results. The start point can be a strong algebraic system, which includes the ordering of the natural or integer numbers. It’s a better option. But the list of axioms has to be quite extensive, which means back to boring. One would like to believe that elementary mathematics is simple but it’s not. And neither is the task of presenting elementary mathematics.

I could leave unclear what is assumed about natural or integer numbers. That’s what usually is done. But I would feel uncomfortable all the time.

Hey, I have a promising idea. In general, of the equivalent systems of axioms some seemingly weak simple ones and some seemingly very strong systems can be highly elegant, for different reasons of course. If you want to prove that a model satisfies the axioms then the weak system of axioms is useful. But when you want to derive new theorems then the strond system gives you a head start. In the theory of smooth manifolds John Milnor superbly selects axioms and definitions which are in between, neither weak nor very strong. It’s a true art. Beautiful and pragmatic at the same time. Let me attempt something similar for the beginning of the Number Theory. I’ll sketch my idea below (in the actual note the axioms will be spelled out in a more detailed way, and will be nicer–no Neg nor + will be mentioned). The axioms will describe only the group of rational integers. The rest–multiplication and ordering–will be defined afterward. Thus indeed this approach is stronger than Peano axioms, but avoids immediate definition of integers as an ordered ring.

DEFINITION   Integland  is an ordered triple   Z  :=  (Z  –  1), which satisfies the following axioms (this are not the ultimate axioms yet, but what the axioms should achieve):

1. Z  is a set;
2. 1 ∈ Z;
3. symbol  –  stands for a binary operation  – : Z² → Z;
4. 1-x ≠ x
5. if  1-x = x-1  then  x=1;
6. (Z + Neg 0)  is an abelian group, where  0 := 1-1,  and the unary operation  Neg,  and binary operation +:

   Neg : Z → Z   and   + : Z² → Z

   are defined as follows:

   Neg(x) := 0-x   and   x+y := x-Neg(y)

   for every  x y ∈ Z;

7. If  A ⊆ Z  is such that:
   • 1 ∈ A
   • ∀_{x y ∈ A}  x-y ∈ A

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   then  A = Z

END of DEFINITION

Now  *  is defined as the unique binary operation such that

• 1*x = x*1 = x
• (a-b)*x = a*x – b*x
• a*x = x*a

for every  a b x ∈ Z.

This is actually an inductive definition of *.