It's not that we rely on such things, but that these things are unavoidable consequences of choices we make that seem to be perfectly reasonable, and result in useful math.

Phrasing in the other way:

As we develop math that we find useful and powerful, we find that we have to make choices. Those choices have consequences, and sometimes as we explore the consequences we find that really strange things happen.

We can go back and make different choices, but in practice we tend to find that no matter what choices we make there are odd and hairy things that result.

The natural way to encode an infinite binary tree into the real numbers is as the Cantor set, an uncountable totally disconnected set.