Please clarify why $|\mathbb{Q}| = |\mathbb{N}|$, but $|\mathbb{R}| > |\mathbb{N}|$? Why do following arguments about $|\mathbb{R}| = |\mathbb{N}|$ are wrong?
Two sets are equinumerous if there is a bijection between their elements. That is, each element of one set is in one-to-one correspondence with an element from the second set.
The cardinality of the set $|A|= |\mathbb{N}|$ if the elements of the set $A$ can be enumerated, that is, there is a bijection $A \longleftrightarrow \mathbb{N}$.
It is known that $|\mathbb{Q}| = |\mathbb{N}|$, since it can be numbered by traversing the diagonals $i=1,\ldots,\infty$.
\begin{equation} \begin{aligned} i=1:\;& \frac{1}{1} \mapsto 1;\\ i=2:\;& \frac{1}{2} \mapsto 2;\; \frac{2}{1} \mapsto 3;\\ i=3:\;& \frac{1}{3} \mapsto 4;\; \frac{3}{1} \mapsto 5;\\ i=4:\;& \frac{1}{4} \mapsto 6;\; \frac{2}{3} \mapsto 7;\; \frac{3}{2} \mapsto 8;\; \frac{4}{1} \mapsto 9;\\ \ldots \end{aligned} \end{equation}
Well, let's denote $\mathbb{R}_{(0,1)}$ is the set of all real numbers on $(0,1)$ and try to enumerate them.
For this, we introduce the concept of depth of a number. A real number $r \in (0,1)$ has depth $i$ if its unique prefix, sufficient for unambiguous recognition of the number, has length $i$. For example, the number $0.0123000 \ldots $ has depth $4$, and the number $0.12000 \ldots $ has depth $2$.
Denote by $\mathbb{R}_{(0,1)}^{i}$ the subset of all numbers from $\mathbb{R}_{(0,1)}$ with depth $i$. Obviously, $\mathbb{R}_{(0,1)}^{i}$ is a finite set for any fixed $i$, and hence it is enumerable set. Therefore, there is bijection $\mathbb{R}_{(0,1)}^{i} \longleftrightarrow \mathbb{N}^i, \; \mathbb{N}^i \subset \mathbb{N}$. Consequently there exists a bijection $\mathbb{R}_{(0,1)} \longleftrightarrow \mathbb{N}$. Hence: $$ |\mathbb{R}_{(0,1)]}| = |\mathbb{N}| $$
Example: \begin{equation} \begin{aligned} i=1:\;& 0.1 \mapsto 1;\; 0.2 \mapsto 2;\; 0.3 \mapsto 3;\; 0.4 \mapsto 4;\; 0.5 \mapsto 5;\; 0.6 \mapsto 6;\; 0.7 \mapsto 7;\; 0.8 \mapsto 8;\; 0.9 \mapsto 9;\\ i=2:\;& 0.01 \mapsto 10;\; 0.02 \mapsto 11;\; 0.03 \mapsto 12, \ldots, 0.99 \mapsto 110; \\ & \ldots \end{aligned} \end{equation} Now let's try to number all real numbers $\mathbb{R}$.
We can extend the definition of depth of a number and assume that depth of $i$ can be not only in one direction (to the right), but also in the other (to the left)
Let's introduce the pair $(i, j)$ where $i$ is the depth of the number to the left, and $j$ is the depth of the number to the right, let's call such a pair the radius of the number. For example, for the number $ \ldots 00012345,810000 \ldots $ we have radius $ (5,2) $, and for the number $ \ldots 000012,9000 \ldots $ we have radius $(2,1)$.
We denote by $\mathbb{R}^{(i, j)}$ the subset of all numbers from $ \mathbb{R}$ with radius $(i, j)$.
$\mathbb{R}^{(i, j)}$ is a finite set for any fixed $ i, j $ and hence enumerable. Therefore, there is bijection $\mathbb {R}^{(i, j)} \longleftrightarrow \mathbb {N}^{(i, j)}, \; \mathbb{N}^{(i, j)} \subset \mathbb{N} $. Hence there is a bijection $\mathbb{R} \longleftrightarrow \mathbb{N} $.
Hence: $$ | \mathbb{R} | = | \mathbb{N} | $$ What is wrong.
Let's call our finite set $\mathbb{N}^{i}$ (or $ \mathbb{N}^{(i, j)}$) the package.
In case of proof, $ | \mathbb{Q} | = | \mathbb{N} | $ our package is the diagonal elements, the number of which in each new package increases by $ 1 $
In the case of the statement that $ | \mathbb{R} | = | \mathbb{N} | $ our package is all possible numbers with depth $ i $ and the size of the next package increases not by $ 1 $, but times.
But does this limit the construction of a bijection? What is the fundamental difference? Indeed, in both cases we have a union of an infinite sequence of disjoint finite sets.
Why is it considered that $ | \mathbb{Q} | = | \mathbb{N} | $ is true, but $ | \mathbb{R} | = | \mathbb{N} | $ is wrong?
P.S. If you are asking what is the depth of $1/3$? What is the depth of the integer $33333(3)$?
