Given a set S, with cardinality $\kappa$, let F be a family of subsets of S, such that if A and B are any two members of F then either A $\subset$ B or B $\subset$ A.
Question: Can you always find F, such that card(F) = $2^\kappa$ ?
Motivation: I recently proved this is true if S = $\mathbb{N}$.
However, my proof is dependent on very specific properties of $\mathbb{R}$ and $\mathbb{Q}$, so I was wondering if the result could be generalized.
Proof: Let f be a bijection from $\mathbb{Q} \rightarrow \mathbb{N}$ be a bijection.
For every real number r, define Q(r) to be the set of rationals < r.
Now define corresponding subsets of $\mathbb{N}$, N(r) = f(Q(r)).
Clearly, the F = {N(r)} satisfies the criterion and card(F) = $2^{\aleph_0}$
QED
