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.

  • $\begingroup$ The p-adics are equipollent to the reals. Another way of looking at them at least. $\endgroup$ Jan 25 '12 at 4:14
  • $\begingroup$ Although I posted an answer (hoping it's not an overkill) I don't quite understand the question. Do you want a visualization of real numbers? Because in mathematical folklore Russian Matryoshka doll analogy is prevalent. As for transfinite number visualization, Rudy Rucker in Infinity and the Mind has a sketch of ziggurat of transfinite numbers. $\endgroup$ Jan 25 '12 at 4:28
  • $\begingroup$ I'm not entirely sure what you're looking for. There are many ways to visualize the real numbers, but it sounds like you're looking for one that has a certain property with the Cantor set. Can you elaborate a bit on that (if you get notifications and are still interested)? $\endgroup$
    – Mark S.
    Dec 1 '13 at 23:41

Stephen Brooks has a website that has a way to visualize Transfinite Number Line

enter image description here enter image description here enter image description here

  • $\begingroup$ :May I ask you what does "$ω$" stand for in the pictures above? Why do we see nothing between the Natural numbers? Thanks. $\endgroup$
    – Mikasa
    Jan 25 '12 at 16:23
  • 1
    $\begingroup$ @BabakSorouh It stands for ordinal number. Yes, it does not show the numbers between N but it was to answer the part about "totally 'radically' different way[s]" of looking at the real line. $\endgroup$ Jan 25 '12 at 16:35

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