I sense that it is not possible.
But, I have been unable to demonstrate precisely why it is not possible.
I have tried something along the lines of:
Consider two real numbers, $r_1, r_2$ such that $r_1 < r_2.$ The real number $r_1$ corresponds to an open interval $i_1$ whose lower bound is strictly less than that of $i_2,$ which corresponds to $r_2.$ I then tried to show that there are infinitely many real numbers between $r_1$ and $r_2,$ but not infinitely many open intervals between $i_1$ and $i_2.$ But, it seems that there are.
Is this about the right way to approach the probably, that is, to understand the real numbers to be in bijection with the uncountable set of open and disjoint intervals, and then show a contradiction?
Guidance will be appreciated. A given answer will purloin the possibility that I may come upon the insight myself, which, the realization of that possibility being a joyful experience, will leave me a bit grave!
Answer
It sounds like your attempt was implicitly assuming two things:
"Uncountable" means "In bijection with $\mathbb{R}$."
The putative bijection needed to be "order-preserving" in the sense that if $r_1
Both of these are unjustified. In particular, the first one in this context is essentially the continuum hypothesis, which is known to be neither provable nor disprovable from the usual axioms of set theory!
Instead, you're going to need to think a bit more geometrically. The idea, essentially, is that there isn't enough "room" to fit uncountably many open intervals into $\mathbb{R}$ without some overlap. I think a good hint at this point is to observe that open intervals are "filled in" in a sense: any dense subset of $\mathbb{R}$ intersects every (nonempty) open interval. So:
Can you think of a dense set of reals which is "small" in some sense having to do with cardinality?
The point is that disjoint open sets overlap a dense set in different ways (they can't overlap at the same point), so if we have a "small" dense set we have in some sense a limit to how many disjoint open intervals we can get.
This should get you started (and hopefully doesn't give away the answer); leave a comment if you want me to provide further hints.
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