Take any $a$ in $(0,1)$ (the real numbers between $0$ and $1$). Consider the decimal expansion $a=0.d_{1}d_{2}d_{3}...$ (where each $d_{i}\in\{0,1,...,9\}$, and the expansion does not end in a string
of infinitely many $9$'s (for example $0.122$ is not written as $0.121999999...$.)). Let $\{p_{1},p_{2},p_{3},...\}=\{2,3,5,...\}$ be the prime numbers. Define the subset $B_{a}=\{2^{d_{1}},2^{d_{1}}3^{d_{2}},2^{d_{1}}3^{d_{2}}5^{d_{3}},...\}$ of $N$. Then if $a_{1}$ is not equal to $a_{2}$, $B_{a_{1}}$ is different from $B_{a_{2}}$ (using unique prime factorization). So the collection $C=\{B_{a}:a\in(0,1)\}$ is uncountable. Moreover, if $a_{1}$ is different from $a_{2}$, then there is a smallest $k$
such that $a_{1}$ and $a_{2}$ differ in the $k$-th decimal. So the intersection between $B_{a_{1}}$ and $B_{a_{2}}$ is $\{2^{d_{1}},2^{d_{1}}3^{d_{2}},2^{d_{1}}3^{d_{2}}5^{d_{3}}...p_{k-1}^{d_{k-1}}\}$, a finite set (or empty if $k=1$).