A [recent question](https://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
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[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.



Cardinality of a certain set of distinct subsets of $\mathbb{N}$
A simpler construction, which yields pairwise incomparable sets (but not almost disjoint sets): 


For any subset $A\subseteq \mathbb N$, let $X\_A:= \{ 2n: n\in A\}$, and 
let $Y\_A:= \{ 2n+1: n\notin A\}$, and let $Z\_A:= X\_A \cup Y\_A$. 


Then the family of all $Z\_A$ has size continuum, and $Z\_A \subseteq Z\_B$
implies both 


* $A\subseteq B$ (because of $X\_A\subseteq X\_B$)
* and also $B \subseteq A$ (because $Y\_A\subseteq Y\_B$),


hence $A=B$. So the $Z\_A$ are pairwise incomparable. 


(A similar construction works also for larger cardinalities; a set of size $\kappa$ has $2^\kappa$ many pairwise incomparable subsets; here I use a bijection between $\kappa$ and $\kappa\times 2$.)


