The same 5 rules of the Peano Axioms repeat over and over again as we develop elementary arithmetic of whole numbers. The 5 rules apply not just to the natural numbers, but can be restated to apply to the head sets as well.
So the sequence 0,1,2,3,… has the 5 rules.
- Start node.
- Out of each base node a single arrow consisting of a base node and tip node.
- The start node is not a tip of any arrow.
- No node is the tip of two arrows.
- If a path contains the start node and the tips it points to, then it contains all the nodes.
The sequence of head sets also follows these 5 rules
ie the head sets of natural numbers.
So do the sets
ie the head sets of the counting numbers.
The 5 rules apply to the unit fractions, i.e. to the sequence
They apply to any sequence 0/n, 1/n, 2/n
They apply to any sequence m/1, m/2, m/3,…
Showing this teaches meaning to students. This has not been pointed out by math profs or math literature or math education. This is because they don’t do a good job explaining Peano Axioms. This is because they don’t fully understand it.
Math textbooks put the easy stuff in the problems. They put the think stuff in the problems. So the textbook writers don’t write the explanations of this. You can’t improve what you don’t write. Which is the result of the practice of putting a lot of steps and easy stuff into math problems instead of the text. The textbook writers never write the explanations and never improve them. So they don’t get developed. This is why the above points have never been articulated before.