Find a poset with infinitely many infinite chains and infinitely many infinite antichains.

Question

I need to find some poset with infinitely many infinite chains and infinitely many infinite antichains. I guess it can be something with real numbers or prime numbers. But I can't find any relation like this between them.

Answer 1

HINT: Find a suitable partial order on $\Bbb Z\times\Bbb Z$. I’ve left an additional hint in the spoiler-protected block below.

Arrange it so that the chains are the sets $\{n\}\times\Bbb Z$ for $n\in\Bbb Z$.

Answer 2

Hint: Consider $\mathcal{P}(\mathbb{N})$, partially ordered by inclusion