  The cardinality of real numbers depends on the chosen axiomatic system and
specific models of set theory. Under the ZF framework, real numbers are
nondenumerable, but their exact cardinality remains debated due to possible
extensions of the ZF axioms. We explored this issue by analyzing sets within
the ZFC model and Wang's $\Sigma$ model. Set $S'$, within the ZFC framework,
includes all pairs of algebraic numbers with a transcendental number inserted
between each pair by a function $f$, which inserts an arbitrarily chosen
transcendental number into every interval bounded by algebraic endpoints. This
approach enables the representation of any number using these intervals. By
employing the AC, one can identify intervals from any Cauchy sequence within
$S'$. As each interval contains at least one number due to the function $f$,
the intersections of these intervals contain numbers defined by these
sequences, making a complete and countable set $S'$.
  The second set, created by inserting specifically defined transcendental
numbers between algebraic numbers, starts with the sequence of algebraic
numbers as the 0-th layer in Wang's $\Sigma$ model. A diagonal procedure
applied to that layer generates a set of transcendental numbers which, combined
with the 0-th layer, forms the $\Sigma_1$ layer. Continuing this recursive
process results in a set equivalent to $S'$. This suggests that all
constructive real numbers are countable, challenging the traditional diagonal
argument. The AC is crucial for selecting transcendental numbers and
identifying intervals and sequences, making the set $S'$. This might resonate
with constructivists who oppose the AC. However, further analysis revealed
conflicts between proofs of non-denumerability and the Nested Interval
Property, indicating a need to reevaluate the countability of real numbers.
