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I know how to prove real number is uncountable. But since both real number and natural number are linear ordered, it feels like there is some one to one relation between those two. Like the infinite hotel story, since they are both ordered, we can move them in the hotel one by one. But this is obviously wrong, I just can not figure out why. Thanks.

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    take a real number $r\in \mathbb{R}$. Which real number is next? – David Diaz Jul 07 '18 at 04:21
  • So it is because real number r do not has any successor, then there is no one to one relation? But I not quite understand why it can be no successor but still linear ordered? –  Jul 07 '18 at 04:34
  • To show that something is possible, all you need is one example. The set of real numbers, modeled as the real number line, with left-to-right ordering (i.e.. $a < b$ if $a$ is to the left of $b$), is an example of a set which is linearly ordered, but where no element has an immediate successor. – quasi Jul 07 '18 at 04:39
  • I know real number has no immediate successor is a fact, but I am not sure how to prove it. And it feels counterintuitive, since if one can not define what followed with a real number, then what ordered even means in real number. –  Jul 07 '18 at 04:44
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    "What ordered even means" That if you take two real numbers, $x,y$, then you have exactly one of the three following: either $x<y$ or you have $x=y$ or you have $y<x$. "Real number has no immediate successor, but I am not sure how to prove it" Consider $0$ and suppose that a proposed "successor" to $0$ that we'll call $\epsilon$ exists. But then $0<\frac{\epsilon}{2}<\epsilon$ and so $\frac{\epsilon}{2}$ is a smaller number than $\epsilon$ which should have been a better candidate for a successor than $\epsilon$ is, a contradiction. – JMoravitz Jul 07 '18 at 04:47
  • Thanks, I understand how to prove the no successor part. But I still not quite understand the meaning of ordered part. I feel like I may misunderstand the what "linear" means here. I thought "linear" means all the number can be arrange from smallest to largest in a line (although there is no the smallest and largest in real number), and every element of real number has a position in the line. Then what ordered or greater than means that a real number b is behind real number in the line, then b > a. So since every one has a position in the line, then no successor and still ordered feel not –  Jul 07 '18 at 04:56
  • feel not compatible with each other. But is the linear order just only means every two real number can be compared, and there is no such line exist? But if is so, the Cartesian coordinate feels unnatural. –  Jul 07 '18 at 04:58
  • You're mostly struggling with what a real number is. Maybe you should study a text where they are constructed from the rational numbers? The rational numbers already are densely linearly ordered (no successors/predecessors) and the reals can be seen as a completion of the rationals in the order sense, filling in the gaps in the rationals. – Henno Brandsma Jul 07 '18 at 04:59
  • So, do you understand how all the rationals are linearly ordered? That is a still a countable example, but the completion process makes it uncountable. – Henno Brandsma Jul 07 '18 at 05:01
  • Yes, I think I do, as linear ordered only requires partial order and every two elements are comparable, and rationals meet those two requirement. –  Jul 07 '18 at 05:03
  • But I not quite understand the completion part, since if we add something extra to a countable infinite set, the set can still be countable. Why we add something to rationals, it must be uncountable? –  Jul 07 '18 at 05:07
  • Because we have to add the supremums of all sets of rationals that do not have a supremum yet. And a countable set has uncountably many subsets. – Henno Brandsma Jul 07 '18 at 05:09
  • If we add a countable number of things to a countably infinite set, the set will still be countable. If we add an uncountable number of things to a countably infinite set, it becomes uncountable. Here, we added an uncountably infinite number of numbers, namely the irrational numbers. If we decided to include countably many things to the rationals, it would still be countable. – JMoravitz Jul 07 '18 at 05:12
  • If you study the construction of real numbers from Dedekind cuts in the rationals, we can prove we get a linear order, when we know we have a linear order on the rationals. IMHO everyone studying university-level maths should have seen a construction of the reals from the rationals. Even if only to build intuition. – Henno Brandsma Jul 07 '18 at 05:13
  • "But since both real number and natural number are linear ordered, it feels like there is some one to one relation between those two." Why? Why would you think that? "But I not quite understand why it can be no successor but still linear ordered? " Why should being linear ordered have anything to do with successors? "I thought "linear" means all the number can be arrange from smallest to largest in a line" Ah! Now we're getting somewhere. You can't lay them out one by one, drop by drop, you lay them out in one effing stroke of a gormongeous dripping paint roller. – fleablood Jul 07 '18 at 05:21
  • A not very relevant question just came to me. Can we prove irrational is uncountable no by rational is countable, and real is uncountable, then uncountable minus countable is uncountable, since then "we add an uncountable number of things to a countably infinite set, it becomes uncountable" seems a little circular. –  Jul 07 '18 at 05:25
  • Thanks. @fleablood, the metaphor really help. –  Jul 07 '18 at 05:31
  • Id like to note that this question is different, and more simply resolved than the supposed duplicate. The countability or uncountability of the rationals or reals is an idea that cost Cantor his reputation among his advisors and peers. THIS question can be resolved by much simpler means – David Diaz Jul 07 '18 at 20:26

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