The answer is the cardinality of natural numbers. But I don't understand what is wrong with this argument :-
Every finite matrix is going to have a finite number of rows and columns. So, the number of matrices with a finite size (m,n) would be 2^mn) and since the range of m and n is natural numbers this would be = 2^NN Wich is equal to 2^N which is real numbers cardinality. What is wrong with this logic?
The problem is that you are never actually in a situation where you have infinitely many possibilities for a 0 or 1, which would lead to your $2^\mathbb{N}$ (or rather, $2^{\aleph_0}$). There are infinitely many natural numbers, but each matrix only has a finite number of entries.
Instead, think about this: the set of all finite 0-1 matrices consists of all such matrices of size 1x1, or of size 1x2, or of size 2x1, or of size 1x3, and so on.
Then, how can you express the whole set in terms of these sets of matrices with a particular size, and then what can you say about the sizes of all these sets?