Constructive vs computable real numbers I find it confusing that all of the following statements are true :

The computable real numbers are countable. $-\hspace{-3pt}-$ Alan Turing, &quot;On Computable Numbers, with an Application to the Entscheidungsproblem&quot;
In constructive analysis, the real numbers are uncountable. $-\hspace{-3pt}-$ Everett Bishop, Foundations of Constructive Analysis
&quot;every mathematical statement [in constructive analysis] ultimately expresses the fact that if we perform certain computations within the set of positive integers, we shall get certain results&quot; $-\hspace{-3pt}-$ Ibid.

Perhaps I am misunderstanding something.
I suppose I really have two questions. In constructive analysis :

Why isn't every real number computable?
How is it possible to construct an uncountable set?
It is consistent with the constructive analysis that every real is computable. (In fact, it is consistent with intuitionistic ZF $\mathsf{IZF}$.) This follows from that the consistency of Church's thesis, which claims every total function from $\mathbb{N}$ to $\mathbb{N}$ is computable.
However, it does not mean constructive analysis can prove every real is computable. We know that classical analysis is a superset of constructive analysis (as a theory), that is, every statement that is provable from the constructive analysis is also a theorem of the classical analysis. And classical mathematics proves not every real is computable. Hence it would be accurate to say whether every real is computable is independent of constructive analysis.
To add some comment, I think it does not mean Bishop is simply wrong. Bishop's constructive analysis is minimal in the sense that it is contained in classical analysis, Brouwer's intuitionistic mathematics, and Recursive constructive analysis (also known as Russian constructivism.) The latter one reflects the behavior of recursive mathematics, so Bishop's claim would be true in that sense.

$\mathbb{R}$ is uncountable in the sense that there is no bijection between $\mathbb{N}$ and $\mathbb{R}$. The proof is available from Bishop's Constructive analysis. (Theorem 2.19 of Bishop and Bridges.) Here is a rough proof:

Theorem. There is no bijection from $\mathbb{N}$ to an interval $[x_0,y_0]=\{z\in\mathbb{R}\mid x_0\le z\le y_0\}$.
(Caution: $x\le y$ is not ($x&lt; y$ or $x=y$.))
Proof. The proof uses diagonalization argument. Assume that $f:\mathbb{N}\to\mathbb{R}$ is a function. We will find a Cauchy real $x\in [x_0,y_0]$ such that $f(n)\neq x$ for all $n$.
We will construct sequences of natural numbers $(x_n)$ and $(y_n)$ recursively such that

$x_0\le x_n\le x_m&lt;y_m\le y_n\le y_0$ if $1\le n\le m$,
$x_n&gt;f(n)$ or $y_n&lt;f(n)$, and
$y_n-x_n&lt;1/n$.

Assume that $x_i$ and $y_i$ are given for all $i&lt;n$.
Then we have either $f(n)&gt;x_{n-1}$ or $f(n)&lt;y_{n-1}$. (This follows from the following constructively valid theorem: if $x&lt;y$ are reals, then either $z&lt;y$ or $x&lt;z$.)
Assume that we have $f(n)&gt;x_{n-1}$. (The remaining case is analogous.)
Choose a rational number $x_n$ and $y_n$ such that $x_{n-1}&lt;x_n&lt;\min(a_n,y_{n-1})$ and $x_n&lt;y_n&lt;\min(a_n,y_{n-1},x_n+1/n)$. Then the mentioned inequalities are satisfied.
Hence $(x_n)$ forms a Cauchy sequence of rational numbers. Let $x$ be a limit of $(x_n)$, then $x$ satisfies the desired properties.

Note that the above proof makes use of the countable choice that Bishop accepted. As far as I know, it is open whether $\mathbb{R}$ is countable or not without the countable choice. Also, note that $\mathbb{R}$ can be subcountable, i.e. $\mathbb{R}$ can be an image of a subset of $\mathbb{N}$. (Unfortunately, I forgot whether the subcountability of $\mathbb{R}$ is compatible with $\mathsf{IZF}$, although I believe it is true.)
1.Because there is at least one real number that is not computable.
2.
Step 1 : 0.0 , 0.1
Step 2 : 0.00 , 0.01 , 0.10 , 0.11
Step 3 : 0.000 , 0.001 , 0.010 , 0.100 , 0.011 , 0.101 , 0.110  , 0.111
step n : 0.(all combinations 1,0 of length n)
continue above indefinitely in binary to get all values in the interval 0,1.