"Count-ability" of the infinite and cardinals

[Ross A. Finlayson]

Cantor's theorems are quite most simple and for
each of the "original", "first", "anti-diagonal", "powerset",
theorems of Cantor for uncountability, that there only
exists one function "counting the real numbers" is from
that there exists a model of real numbers with countable
additivity if each measure zero in the field's real numbers,
that is also exactly (and, only) measure 1.0.

Which is "most simple".

Then, that the rationals are huge, speaks to the m-w proof,
where besides "nested intervals are dense in rationals, too".

So, then for the powerset and "ubiquitous ordinals", is why
it is all simple and contrived for foundations why then the
measure-theoretic takes effect from this simpler rule, than
the usual algebra's arithmetic's field's completion of the ordered
field (also usually the rationals), then defining "LUB" then
"measure 1.0".

Thus, the "anti-diagonal" and countability of a [0,1] by
the Equivalency Function (a "super-" or "non-" or "sub-"
Cartesian function), is neat and for proper fundamentals
for a mathematics above geometry's integer and real
arithmetics, all modern mathematics in set theory.

Sweep: no more to know.