I am only a first semester undergraduate (of mathematics) so I would appreciate the answers to not be too complicated.
Also English is not my first language, so my explanations might be a little off.
While trying to prove that the Powerset $\mathcal{P}(\mathbb{N})$ is uncountable I ran into something that I wasn't able to understand.
So, I define the set
$$
\mathbb{N}_n=\{\{a_1,a_2,\ldots,a_n\}:a_i\in \mathbb{N} \; \forall i \text{ and } a_i \neq a_j \; \forall i \neq j\} \text{ for } n \in \mathbb{N}
$$
basically the set goes like this
$$
\mathbb{N}_0=\{ \emptyset \} \\
\mathbb{N}_1=\{ \{1\},\{2\},\{3\},\ldots \} \\
\mathbb{N}_2=\{ \{1,2\},\{1,3\},\{2,3\},\{1,4\},\ldots \} \\
\vdots
$$
and we can conclude that
$$
\text{if } \mathbb{N}_i \cap \mathbb{N}_j = \emptyset \text{ then } i \neq j \\
\bigcup\limits_{i \in \mathbb{N} \cup \{0\}} \mathbb{N}_i = \mathcal{P}(\mathbb{N})
$$
Okay, so there are a countable number of sets $\mathbb{N}_i$. Also the set $\mathcal{P}(\mathbb{N})$ is uncountable. Also as briefly mentioned in yesterdays lecture (at my university), a countably infinite union of countable sets is a countable set. From this I have concluded that:
$
\text{There exists an } i \in \mathbb{N} \text{ such that the set } \mathbb{N}_i \text{ is uncountable.}
$
Now, there is the possibility of a mistake that I have made in a previous statement. But in the case that I haven't, this last statement seems very counter intuitive to me. I have also tried proving / disproving it but failed.
My question is where have I either made a mistake, or if I haven't how do I prove or disprove my final statement.
Best Answer
Your set is a set of all finite subsets of $\mathbb N$. Notice that none of your sets contains a set of all even numbers: $$\forall (i\in\mathbb N)\ \{ 2k \mid k\in \mathbb N\} \notin \mathbb N_i$$ or any other infinite subset of $\mathbb N$.