A recent question here has convinced me that folks here have a warm heart for the foundations of quantum mechanics, so I decided to ask a question that has been bothering me for a while.

Quantum motivation

Hardy has proved a theorem saying the the cardinality of the ontic space $\Lambda$ must always be infinite. As Spekkens put it more clearly, his proof works by making a injection of the set of pure quantum states into the set of distinct subsets of $\Lambda$. Since the set of pure states is continuous, it follows that $\Lambda$ must be infinite.

But the injection isn't exactly onto $\mathcal{P}(\Lambda)$, but rather onto a set $D$ of distinct subsets of $\Lambda$ such that that for no $A,A'\in D$ is it true that $A \subset A'$. My question is then: is this additional restriction enough to show that $\Lambda$ must be continuous? Or is there a countable $\Lambda$ such that $D$ is uncountable?

Quantum-free question

Let $D$ be a set of distinct subsets of $\mathbb{N}$ such that for no $A,A'\in D$ is it true that $A \subset A'$. What is the maximal cardinality of such a $D$?

This question seems simple enough, but I haven't been able to answer it.