The pitiable arithmetic of one, two, many...

Occasionally one will find accounts of hunter-gatherers whose numbering system goes as one, two, many.  Such accounts usually have just a whiff of superiority about them.

Nevertheless, one, two, many, provides a small but workable arithmetic, especially if augmented with nothing and don't know, e.g. :

${\displaystyle 1+1 = 2,}$















${\displaystyle 2-1 = 1,}$









The funny thing is that we ultranumerate moderns have something that behaves a lot like many.  We call this thing infinity.  So, for instance,

${\displaystyle 1+\infty = \infty ,}$

${\displaystyle 2+\infty = \infty ,}$

${\displaystyle \infty +\infty = \infty ,}$



In particular, ${\displaystyle \infty }$ doesn't scale :  ${\displaystyle n \infty = \infty .}$   nothing and ${\displaystyle 0}$ don't scale either, but there the behavior is just what we would expect.  With ${\displaystyle \infty }$, as with many, the behavior seems much less legitimate :  do we really expect any large thing to be the same as twice itself?

Why not just invent a well-behaved, i.e. scalable, infinity, or at least a unit of infinityness?  Let us invent one, and call it ${\displaystyle \Omega }$.  We require that, for instance, ${\displaystyle 2\Omega >\Omega .}$  In fact, we treat ${\displaystyle \Omega }$ as a new kind of number that is larger than any real number, but is otherwise as well-behaved as we can get it to be.

It is amusing to play with this quantity.  But for real fun, we need to look at quantities in base ${\displaystyle \Omega }$.  For that will take us to infinitesimals and calculus.