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I understand that Cantor set isn't countable and I accept the proof also.

But, what if we count the elements of the set like the following?

1, 0, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, 1/27, 2/27, 7/27, 8/27, 19/27, 20/27, 25/27, 26/27

It might help to see the image in this link, to see how the sequence I wrote above goes.

http://en.wikipedia.org/wiki/Cantor_set

Basically every time you cut off the middle part from each bar, you're counting the endpoints from the left, excluding the ones you've already counted.

Anyway, to my intuition, this sequence can be matched with positive integers, which means it's countable.

What am I getting wrong here?

I came across this set recently so I don't have much understanding about it. It would be great if some of you can give me some insight. Thank you.