I have a question that relates to this question about Venn diagrams. Has anyone shown that all Venn diagrams can (theoretically) be constructed?
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3$\begingroup$ Can you please clarify what you mean by 'all Ven diagrams' and 'theoretically' and 'be constructed' mean? Without precise meaning for these the question is too vague. $\endgroup$– Ittay WeissJan 24, 2013 at 1:56
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1$\begingroup$ Would you allow uncountable families of sets? These don't admit Venn diagrams in any reasonable sense. $\endgroup$– Trevor WilsonJan 24, 2013 at 2:06
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1 Answer
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There is a recursive construction for Venn diagrams (see the Wikipedia page for Venn Diagram), so by induction there is a Venn diagram for any finite $n$. For $n\approx\omega$, it is more complicated, since $2^\omega\approx\mathbb R$, but I think it can be done. But if $n\succ\omega$, then $2^n\succ\mathbb R^2$, so it can definitely not be done on a plane (because there simply aren't enough points on the plane to do the job).
answered Jan 24, 2013 at 2:19