# Is this set the Power Set of Natural Numbers?

I have a school problem and one of the affirmations I have to prove about it seems -to me- contradictory to what my intuition says.

Let $$Q_n:\{m \in \mathbb{N}: m>n\}$$ and $$P_n$$ the family of all subsets of $$\{1,...,n\}$$. Let us define

$$L_n=\{L:L \in P_n \enspace or \enspace L=P \cup Q_n, P \in P_n\}.$$

My intuition tells me that $$\cup_{n=1}^{\infty}L_n=P[\mathbb{N}]$$. Where $$P[\mathbb{N}]$$ is the Power Set of Natural Numbers.

• Yes, thank you. I already corrected that. I am writing from my mobile and it's a bit hard. Commented Sep 9, 2019 at 18:59
• The set of even natural numbers is an element of the power set. Is this an element of your set? Commented Sep 9, 2019 at 19:01
• Your set includes finite subsets, and cofinite subsets; does it include infinite sets with infinite complements? Commented Sep 9, 2019 at 19:05
• This is what I don't completely understand. $P_n$, by definition, is the Power Set of the finite set $\{1,...,n\}$. By $\cup_{n=1}^\infty P_n$ doesn't this become the Power Set of Natural numbers by gathering all of the subsets of the set \{1,...,n\} when $n \rightarrow \infty$ that leads to $\mathbb{N}$? Commented Sep 9, 2019 at 19:12
• No, it doesn't, because for every $n$ each element of $P_n$ is always finite. In your union you will have the set of the first $n$ even natural numbers for any finite value of $n$ that you wish, however large., but you will never have the set of all even natural numbers. Commented Sep 9, 2019 at 19:16

For every $$n$$, every set in $$L_n$$ will either be finite or will include every natural number greater than $$n$$.
Every set then in $$\bigcup\limits_{n=1}^\infty L_n$$ will either be finite or will have some value of $$n$$ for which every natural number greater than $$n$$ appears. Remember... $$A\in \bigcup\limits_{n=1}^\infty L_n$$ is true if and only if there exists some $$n$$ for which $$A\in L_n$$.
As such, the set of even natural numbers is not in your set as it is neither finite nor has a value $$n$$ for which every number greater than $$n$$ appears in it.