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. :

`\qquad\qquad\qquad 1 + 1  =  2,`

`\qquad\qquad\qquad 1 + 2  =  "many,"`

`\qquad\qquad\qquad 1 + "many"  =  "many,"`

`\qquad\qquad\qquad 2 + 1  =  "many,"`

`\qquad\qquad\qquad 2 + 2  =  "many,"`

`\qquad\qquad\qquad 2 + "many"  =  "many,"`

`\qquad\qquad\qquad "many" + "many"  =  "many,"`

`\qquad\qquad\qquad 1 - 1  =  "nothing,"`

`\qquad\qquad\qquad 2 - 1  =  1,`

`\qquad\qquad\qquad 2 - 2  =  "nothing,"`

`\qquad\qquad\qquad "many" - 1  =  "don't know,"`

`\qquad\qquad\qquad "many" - 2  =  "don't know,"`

`\qquad\qquad\qquad "many" - "many"  =  "don't know."`

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

`\qquad\qquad\qquad 1 + oo  =  oo,`

`\qquad\qquad\qquad 2 + oo  =  oo,`

`\qquad\qquad\qquad oo + oo  =  oo,`

`\qquad\qquad\qquad oo - oo  =  "don't know."`

In particular, `oo` doesn't scale :  `n  oo  =  oo.`   nothing and `0` don't scale either, but there the behavior is just what we would expect.  With `oo`, 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 `\Omega`.  We require that, for instance, `2 \Omega > \Omega.`  In fact, we treat `\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 `\Omega`.  For that will take us to infinitesimals and calculus.