Given 1) the power set of the natural numbers.
And 2) the set $\{T\subseteq\mathbb{N}\ |\ T$ is finite or $\mathbb{N}\setminus T $ is finite}.
We know that in 2) there are all finite subsets of the natural numbers. For every infinite subset $T$, there is the finite complement $\mathbb{N}\setminus T$, so now we have all infinite subsets also in the set 2).
But these are all the subsets of the natural numbers, so the power set.
I am asking because I got an assignment in which I have to prove that set 2) is not complete. But since 1) and 2) are equal and the power set is complete, 2) must be complete, too.
So, the only way 2) can be not complete is, when 2) and 1) are not equal.
Am I missing something here?
Yes, consider the even numbers. Neither them nor the complement is finite!