Oh, that's about what's "constructible", what's constructible is countable.
Also called "define-able", "discern-ible", running out all the finite words,
the usual problem is "not including any infinite words", or, "not including
all these infinite words".
Each infinite sequence is, ..., "define-able", by the value of the elements of
the sequence, though then in real numbers that gets to dual representation.
Here it's instead that two copies of the integers essentially make a square,
then that there's only one antidiagonal, and no other everywhere-non-diagonal,
and it so happens that the one antidiagonal is always at the end of the list.
This is due the properties of the function and linearity and so on, as a limit of
functions of integers and it's n/d for, numerator and denominator, where the
denominator goes to infinity.
Can you say exactly where in your list the antidiagonal or some everywhere-non-diagonal
is? In a thory where all functions are Cartesian, Virgil will show exists his algorithm,
that as well-defined as the sequences are, is the antidiagonal.
As far as these being "real numbers R[0,1]" it results that it's a different model
(in model theory for set theory) than the usual model "R the complete ordered
field, and a proper subset R[0,1], bounded by 0 and 1", instead it's "R these iota-values
according to function theory that fills [0,1[, has a total natural well orderingsame as
the integers, and is a continuous domain, is countable, and is unique among functions
that are 1-1 and onto a continuous domain". I.e. a different model of a continuous
domain for set theory, function theory, and what results all else the rest of mathematics,
has all the properties of a continuous domain its elements, then with regards to a
constructivist's "rather restricted transfer principle" or "Schmieden and Laugwitz,
who are constructivists and like whole things countable".
Then these days "metrizing ultrafilter", "Schwarz function support", complementary topoi,
about Vitali and Hausdorff geometers and Banach and Tarski algebraists, it just results
that after apologetics and definitions in function theory and topology, that it's very
simple again that "R[0,1] is as much a whole set clock arithmetic according to any
granularity of for example time", while, "R[0,1] is only a subset of the complete ordered
field reduced to the unit interval".
That otherwise all those things have pretty directly ways to apply arguments, ...,
"either well-defined and founded by multiple models of continuous domains,
or, inconsistent with fundamental ordinary relations, set theory".
Then, after scale in b^p and numerical precision in the algebraic according to
arithmetic coding, all words, each define-able and construct-ible, it's very simple
again and much better for mathematics in terms of that relevant foundations for
all sorts analysis have much more brief and closed derivations, that establish most
all usual definition.
You might axiomatize "there is a big infinity and confoundingly it exhausts
these regions in sequences as well as a sequence little infinity exhausts the
sequence", you might axiomatize that: but then it just results that you have
your own theory, and to say anything at all in terms of theories that are otherwise
totally blind to each other, in model theory it's in terms of transfer principle,
transfer of valid inference about properties, that then going about building
the theories soundly together, is what's called foundations of mathematics,
that alone are instead planks or platforms, of mathematics: here though
that involves solving that the other theory has the opposite conclusion as
a more-than-less direct, inference.
To solve the theories together I arrived at there are two models of continuous
domains at least, then it involves all technical philosophy if though to reduce
to extra-ordinary set theory: that results for that besides the (four or five, ...)
proofs of "uncountability of the reals, ...", is about the one proof "uncountability
of the powerset of integers", that to solve the paradoxes, is also an exercise
in showing that ordinals the objects fulfill showing the exponentiation as increment.
So, there are four or five "Cantor's proofs of uncountability of reals" like antidiagonal
argument and nested intervals, then variously "Cantor's proof of uncountability of
powerset", in "a set theory where all models of functions are Cartesian and not all
models of ordinals are compact", in larger theory there's a less-than-Cartesian function
in terms of its space or the support, and ordinals are ubiquitous and make order theory,
Then, that it works out, "it's the same unique counterexample for all those, this
natural/unit equivalency function or after the slate, and for powerset, the modular
and clock", this then I called "sweep" so the function and principle about make it so
that both it's simple and it's foundations and it's all modern foundations.
So, ..., I provide _one_ example.