I was just wondering if there are any diagrams for visualizing subsets of the real numbers, or totally 'radically' different ways of looking at them as a real line? The model of the line relies on order - which is great for intervals, but you can hardly draw the cantor set on there.
This got me thinking - a set with zero measure but same cardinilty as any interval; the image does not represent that 'length' (i.e. intervals) and 'size' (i.e. cardinilty) are well represented on the picture.
Before this seems like a stupid question, I would like to add it isn't impossible to try to visualize infinite sets; for instance, I think of the power set of $ \mathbb N $ as the graph of all functions with domain being the natural numbers and image 0 or 1. Okay, this doesn't say much about it's cardinilty from the diagram but it's trivial to prove the sequences of 0 and 1's are uncountable (similarly you could argue as the Cantor set is homemorphic to the set of sequences you can visualize it similarly, but I'm not so keen on that).
Thanks for the replies sorry if this goes nowhere, but the responses are usually illuminating on all SE threads.
Stephen Brooks has a website that has a way to visualize Transfinite Number Line