[SaintRomuald]

I never understood this example used to explain it, vsauce made a video on the banach tarski theorem.

You make an infinite list of numbers between 0 and 1 chosen at random. Apparently you can make a new number that was never seen in the list before if you pick a digit from each number in the list and add one to it.

Say the list has numbers 0.36285728.. 0.95825597.. 0.47264112.. .. I can make a new number by taking the 3 from the first number the 5 from the second and the 2 from the third num and so on. 0.463.. I never understood

[ColinWright]

You have misunderstood, or misremembered.

You don't take "3" from the first, you consider three, but choose something that's not "3", so the number you are constructing differs from that first number.

Then in the second place you don't take "5", you choose something that's not "5", so the number you are constructing differs from that second number.

And so on. So every time you have a list of real numbers, it cannot contain all real numbers ... you can always construct many, many, many numbers that are not in your list.

But I'm not sure how this is related to what we've been talking about. If you understand what I've said here and are still confused, maybe you can be a bit more specific. If you have not understood what I have said here, perhaps you can ask more specific questions.