Is the set of all possible finite alphabets uncountable? In my opinion it is uncountable since we can think every natural number as a member of one alphabet like $\quad0\in\sum_0 ,1\in \sum_1, ... \quad$ And there are also other symbols than natural numbers in alphabets. So there is no possible 1:1 and onto mapping from set of all alphabets to natural numbers. Thus they cannot be enumerated which means they are uncountable. Is there anything wrong with my approach?
Your approach can essentially be summarized as:


I define an injection between $\mathbb{N}$ and my set $S$.
This injection is not a bijection because some elements of the codomain are left unmapped.
Therefore there exists no bijection between $\mathbb{N}$ and $S$.


Stated like this, it is perhaps clear that the last statement does not follow from the others. An example of where this reasoning would fail is if we let $S = \mathbb{Z}$ and let the injection be the identity function (mapping naturals to themselves). This leaves all the negative integers in the codomain unmapped, but we know that $\mathbb{Z}$ is countable.