Can Venn diagrams resemble Cantor sets with infinitely many sets?

[fourier jr]

if you go down to the section called "do they always exist?" you'll find a venn diagram for 6 sets:

http://www.combinatorics.org/Surveys/ds5/VennWhatEJC.html
would a horizontal cross-section resemble a cantor set, if there were infinitely many sets? it looks like it would vaguely resemble something like that or maybe I'm missing something. (maybe it's an inane & superficial observation anyway) what does a venn diagram look like for infinitely many sets anyway? that site only has diagrams for small finite numbers of sets but it says it should be clear where the next set should go in the above diagram. would be be possible then to add sets recursively to find the venn diagram for n sets? maybe this belongs in the topology subforum...

 

[blkqi]

The diagram is a sort of fractal geometry...something like that. Most fractals resemble the Cantor set in one way or another; the set was a sort of prototype for fractal geometry.