Does $\mathcal{P}(\mathbb{N})$ contain infinite sets?

Question

I know that $\mathcal{P}(\mathbb{N})$ is infinite and uncountable. However, is the power set of the natural numbers considered to contain only finite sets of natural numbers, or infinite ones as well?

Answer 1

Yes.

Let's pick a few.

• Obviously $\mathbb{N}$ itself
• All the even numbers
• All the odd numbers
• All the prime numbers (these are infinite, right?)

So forth.

(I have exhibited a few infinite sets $\in\mathcal{P}(\mathbb{N})$ thus shown the claim: $\exists S\in\mathcal{P}(\mathbb{N})$ with $|S|=|\mathbb{N}|$ which is to say $S$ is countably infinite)