## One is a label

The road to fractions start with the realization that one is a label.  When we realize we can count labels with the Peano Axioms and that these can stand for bins, even variable ones, we have cracked unity.

When we realize one is a label on a bin, we can look inside.  At this moment, we have split the monad.  Then we can associate whole number labels with bins.  By simultaneously associating unity with a bin of every natural number in size, we get all the unit fractions.  This lets us use each of these unit fractions as a base for Peano Axiom counting along the line of that unit fraction.

This is how we do the fractions.  Each denominator is its own Peano Axiom universe as its numerator varies one by one in accordance with the Peano Axioms.  We can apply recursion rules to the numerator and so define addition and multiplication of numerators holding the denominator fixed.

The unit fractions are the bases of the Peano Axiom universes and taken together let us deal with all the fractions.

Fractions are an association ie a function ie a mapping between the different bin size Peano Axiom sets.  Each bin size is a different association to the base line ie the whole number line.  The whole number line is itself a Peano Axiom line as are each separate bin line.