Is the following infinite set countable?

Question

The following is a subset of $\mathcal{P}(\mathbb{N})$, the power set of the naturals. I am struggling to figure out if it is countable or not. Its complement is uncountable so that doesn't help:

• Take all one element sets that include 1 (so just the set $\{1\}$)
• And all two element sets including 2 but not including 1 (which is isomorphic to $\mathbb{N}$)
• All three element sets containing 3 but not containing a 1 or a 2

...

• All n element sets containing n but not containing any of $1,2,...,n-1$

Answer 1

The set of all finite subsets of a countable set is countable, the union of countably many countable sets is countable. So your set is countable.

Answer 2

The set of all finite subsets of $\mathbb{N}$ is countable and this collection is contained there.