I fail to see why the family of sets Xi is countable. if Xi is of cardinality ℵ0, which I totally agree about, then how can a union of a countable family of them which is basically ℵ0×ℵ0 be equal (0,1)?
Though now that I think about it, if the difference is some irrational number then this seems to work, as any set Xi would contain exactly one unique rational number. Now they each have the cardinality of R, and the family has the cardinality of Q. And then it all seems to work. Does that seem right?
I fail to see why the family of sets Xi is countable. if Xi is of cardinality ℵ0, which I totally agree about, then how can a union of a countable family of them which is basically ℵ0×ℵ0 be equal (0,1)?
The number of such sets is specifically uncountable. Each set is of itself countable. Apologies, I’ll fix the OP.
Though now that I think about it, if the difference is some irrational number then this seems to work, as any set Xi would contain exactly one unique rational number. Now they each have the cardinality of R, and the family has the cardinality of Q. And then it all seems to work.
Does that seem right?
That sounds like it also works. I’ve seen the proof both ways and I think I was mixing them together in my head.