In teaching fractions, we often think of a ruler with inches marked on it, and then we divide the inches into fractions. Thus fractions.
This is a physical model which really relies on our perceptions of physical space as physical beings.
The von Neumann Construction is a chain that can’t be divided this way. It is a counting or indexing chain that is unbreakable.
One version of the VNC is
is one version. There are others.
0 = ∅
In this construction, there is no way to divide one into halves or tenths. We have a chain that is unbreakable.
This is the real concept of counting numbers. We could call them indexing numbers. They keep the place in the count.
When fractions are taught by breaking the inch into eighths or tenths or some division, we miss that one, two, three are really counting numbers that can’t be divided or broken. They index location in a chain like the von Neumann chain.
Fractions then have to be taught as operators on natural numbers.
1/2 acts on even numbers.
1/3 acts on multiples of 3.
We then create rational numbers and then we use point sets on a line or plane as a model of physical space. On that basis we divide the inch into eighths.
We can identify the monad as the unit in the von Neumann Construction.
The inch is a measurement of a dense set of points along a line.
The inch can be broken. The monad can not be broken.