Question

Cantor Set:
I think I follow wikipedia (and my calculus teacher's) explanation of the Cantor set up to the point where they replace all the 2's in ternary with 1's and then treat the numbers as binary.
It seems that the uncountability of the set can be established without this (though I'm not quite sure how). But how is "treating it as binary" even valid? It's not a base-change; it seems quite arbitrary.

Answer 1

hmmm... Well, it's really just the construction of the function. In this construction, we are not claiming that the numbers in ternary have any algebraic connection with the values associated to them through the function, instead, we are simply showing that for all elements of the cantor set, there is a method to assign an element from the unit interval, and more importantly, the corollary: for all elements of the unit interval there is a method to assign an element of the cantor set. This proof is special because it tells us what this method is; some proofs do not give a method or in fact cannot have a finite method. But this is beyond us. By method we really just mean a function, and hopefully this has clarified to you what that means.

Answer 2

So.. it's basically making a connection between the ternary set to a set of binary numbers like the set of all even numbers is the same size of all natural numbers?

Answer 3

Yep. Just like that proof.

Answer 4

Thanks!