Partial order on $\langle P(\mathbb{N}), \subseteq\rangle$ contains continuum-size chains and anti-chains

Question

Consider the partially ordered set $\langle P(\mathbb{N}), \subseteq\rangle$.

I am trying to show two things:

(a) There exists a chain $C$ such that $|C|=\mathfrak{c}$.

(b) There exists an anti-chain $A$ such that $|A|=\mathfrak{c}$.

Can you help me with this ? It seems to be fairly hard.

Answer 1

HINT: You can replace $\Bbb N$ with any other countably infinite set $S$: just use a bijection between $S$ and $\Bbb N$ to transfer subsets of $S$ to corresponding subsets of $\Bbb N$.

• For (a) consider the subsets of $\Bbb Q$ that are the left halves of Dedekind cuts. That is, for each $x\in\Bbb R$ let $A_x=\{q\in\Bbb Q:q\le x\}$, and consider the sets $A_x$.

• For (b) consider the almost disjoint families of this answer.