Give a partition of ω I have a question which I'm deeply confused about. I was trying to do some problems my professor gave us so we could practice for exam, one of them says:
Give a partition of  ω in  ω parts, everyone of them of cardinal  ω.
I know that $ ω=\left \{  1,2,3,...,n,n+1,.... \right \}$ , but I thought that |ω|=ω.
Can somebody help me?

There are many ways to do it, since $|\omega\times\omega|=\omega$. Here’s one. Every $n\in\Bbb Z^+$ can be written uniquely in the form $n=2^km$, where $m$ is odd, and you can start by letting $S_k=\{2^km:m\in\omega\text{ is odd}\}$: $S_0$ is the set of odd positive integers, $S_1$ the set of even positive integers that are not divisible by $4$, and so on. That isn’t quite a partition of $\omega$, since it doesn’t cover $0$, so throw $0$ into one of the parts: let $P_0=\{0\}\cup S_0$, say, and $P_k=S_k$ for $k>0$, and $\{P_k:k\in\omega\}$ is then a partition of the kind that you want.
Another way is to use the pairing function, which is a bijection $\varphi:\omega\times\omega\to\omega$, and set $P_k=\varphi[\{k\}\times\omega]$ for each $k\in\omega$.
