A [recent question](http://mathoverflow.net/questions/95537) 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](http://dx.doi.org/10.1016/j.shpsb.2003.12.001) has proved a theorem saying the the cardinality of the ontic space $\Lambda$ must always be infinite. As [Spekkens](http://arxiv.org/abs/quant-ph/0406166) 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.