Let the Symbol $|\ |$ denote cardinality of an set. Is it possible to construct a family $N_i\subset\mathbb{N}$, $i\in\mathbb{N}$, such that:
1- $|N_i|=\infty$,
2- $\mathbb{N}=\bigcup_{i=1}^\infty N_i$,
3- $|N_1\cap N_i|=i$, $\forall i\geq 2$.
Thanks
Edit: I changed 3.