> Why are interval boundaries real-value in a system that cannot actually express real numbers?

They can be represented as a binary tree, that is a nested list or tuples, for instance. The whole interval (1) could be split into (1,0) and (0,1)). Then (1,0) split into ((1,0),0) and ((0,1),0). And (0,1) split into (0,(1,0)) and (0,(0,1) and so on.

I guess his point remains though that if all you need is infinite divisibility then using the rational numbers between 0 and 1 would be sufficient. I take it as that was what was meant and "real" numbers was just shorthand for that.