Apparently dimension is "informally" defined as "the minimum number of coordinates needed to specify any point within it". For example we need at least 3 numbers to describe any point in the 3D space and 4 numbers for 4D, etc.
However, since that $|\mathbb{R}^2| = |\mathbb{R}^3|$, therefore there must exist a bijection between $\mathbb{R}^2$ and $\mathbb{R}^3$, which also implies that 2 numbers are enough to describe all points within the 3D space, which is sort of surprising to me.
If we can use 2 number of coordinates to specify any point in 3D, then we can do the same thing for any other dimensions. This doesn't match the definition of dimension.
Is there anything incorrect in my logic, or is it that the definition of dimension is more complicated than the one I stated above?
Yeah there is a bijective function from $\mathbb{R}^2$ to $\mathbb{R}^3$ and you can say that $|\mathbb{R}^2| = |\mathbb{R}^3|$. However when you talk about dimension the algebraic structure joins the game. $\mathbb{R}^2$ and $\mathbb{R}^3$ are considered as vector spaces over $\mathbb{R}$. To say that they are equivalent you need special kind of bijection $f : \mathbb{R}^2 \rightarrow \mathbb{R}^3 $, such that $f(x + y) = f(x) + f(y)$ and $f(ax) = af(x)$. Unfortunately such bijection does not exist. The thing that breaks your logic is the structure!