The following two facts about rational numbers confuse me a bit:

  1. Rational numbers are ordered but unlike real numbers are not complete. The incompleteness is due to the existence of gaps in the rational numbers. When we say gaps for this ordered objects, doesn't it intuitively mean that there should exist a pair of rational numbers with a gap in between them.

  2. There exist infinitely many rational numbers in between any two rational numbers. Doesn't this intuitively mean there is no gap between any two rational numbers?

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    $\begingroup$ No and no. Do you know what a Cauchy sequence is and how it's used in the definition of completeness? ps -- It "intuitively" means the wrong thing till you clarify your intuitions! Real numbers aren't lined up like bowling balls. $\endgroup$ – user4894 May 1 '17 at 3:07
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    $\begingroup$ @user4894 : You're introducing complications that may be superfluous. Think of Dedekind completeness. $\endgroup$ – Michael Hardy May 1 '17 at 3:08
  • $\begingroup$ @Dylan : "Gap" in this context has a very precise definition. See my answer below. $\endgroup$ – Michael Hardy May 1 '17 at 3:16
  • $\begingroup$ "Gap" is about the worst possible word I could imagine for this. Okay, "ginkle-snail fish" might be worse but "gap" is a close second. There is no measurable space between rational numbers. And we can find rational arbitrarily close to each other. No "gaps" at all. But infinitely many immeasurably small holes. Among all the infinite rational values, there are always measurable values that simply aren't rational. These aren't "gaps". They are just points that are rational. Actually "ginkle-snail fish" are a better word then "gap". There are infinite ginkle-snail fish. $\endgroup$ – fleablood May 1 '17 at 3:49
  • $\begingroup$ @fleablood Yes, I actually agree but this is the word Dedekind uses (en.wikipedia.org/wiki/Dedekind_cut). I have read Dedekind original paper, where he proposes the cut and it seems he might have had this geometrical interpenetration about the gaps. However, with this geometrical intuition the above contradictions might arise. $\endgroup$ – abk May 1 '17 at 4:12

A gap in the set of rational numbers is a pair of non-empty sets $A,B$ of rational numbers such that

  • every rational number is a member of $A$ or of $B$, and
  • every member of $A$ is strictly less than every member of $B$ (and consequently no number can be a member of both), and
  • $A$ has no largest member and $B$ has no smallest member. For every number $a_1\in A$ there is some number $a_2\in A$ such that $a_2>a_1$ and for every number $b_1\in B$ there is sum number $b_2\in B$ such that $b_2< b_1.$

If $A$ is the set of all rational numbers less than $\sqrt 2$ and $B$ is the set of all rational numbers greater than $\sqrt 2,$ that is a gap.

The set of all real numbers, by contrast, has no gaps.

  • $\begingroup$ Thanks. The "gap" defined in your answer is different from the geometrical interpretation of the gap. However, when talking about the incompleteness, isn't it the case we normally mean the geometrical interpretation of the gap. Like the existence of wholes on the number line as given by Eric Wofsey's answer. $\endgroup$ – abk May 1 '17 at 3:24

A "gap" does not refer to a gap between two individual rational numbers. Instead, it refers to a gap between two sets of rational numbers.

For instance, let $A$ be the set of all rational numbers $q$ such that either $q\leq 0$ or $q^2<2$ and let $B$ be the set of all positive rational numbers $q$ such that $q^2>2$. Then every element of $A$ is less than every element of $B$. But there is no rational number that is "right between" $A$ and $B$. That is, there is no rational number $q$ that is greater than or equal to every element of $A$ and less than or equal to every element of $B$. Indeed, you can prove that if such a $q$ existed, then $q^2$ would have to be $2$, but there is no rational number whose square is $2$. So there is a "gap" between the sets $A$ and $B$ in the rational numbers.

For a simpler example, instead of thinking about rational numbers, consider the set of all nonzero real numbers. This set has a "gap" at zero: you have the negative numbers and the positive numbers, but you have no number in between them. But this gap is not between any two individual numbers, since there is no largest negative number or smallest positive number.

More intuitively, imagine you have a number line and you "point to a spot" on the number line. If there is not actually any number where you pointed, then you can think of that as a "gap" in your number line. The business with the sets $A$ and $B$ is just a way of making the idea of "pointing to a spot" mathematically precise while staying within your number system (which can't refer directly to your spot since it has no number there!). Instead of referring to the spot itself, you can refer to the set of all the numbers that are to the left of the spot (that's $A$) and the set of all the numbers that are to the right of the spot (that's $B$). These sets together tell you exactly where your spot should be.

  • $\begingroup$ Thanks. It seems that the intuitive explanation part of your answer suggest the geometrical interpretation of the gap. My understanding is that with this interpretation of the gap, a missing spot means a missing rational number between two rational numbers. $\endgroup$ – abk May 1 '17 at 3:37
  • $\begingroup$ I don't know what you mean by "the geometrical interpretation of the gap"--that term has no standard meaning. $\endgroup$ – Eric Wofsey May 1 '17 at 3:42
  • $\begingroup$ No. There are no two precise extreme rational numbers that the real number lies between. Yes it lies between two rational numbers but those two will NOT be the closest two. There will never be a closest two. $\endgroup$ – fleablood May 1 '17 at 3:54
  • $\begingroup$ @Eric Wofsey Lets interpenetrate a gap as a whole on a number line and a point on the line as a number (we can assign a number to each point of a straight line). Then if there always exists a point in between any two points, this means that there is no gap. Now, if we only assign rationals to each point of straight line and knowing the fact that there are infinity many rationals (points) in between any two rationals (points), we can then say there is no whole on the number line even for rationals. However we know these wholes exist. This is what confuses me. $\endgroup$ – abk May 1 '17 at 4:00
  • $\begingroup$ "Then if there always exists a point in between any two points, this means that there is no gap." Why do you believe this? As I said, the "gaps" we are talking about are between pairs of sets, not between pairs of individual rational numbers. I would suggest thinking carefully about the example I gave in the second paragraph of my answer. $\endgroup$ – Eric Wofsey May 1 '17 at 4:04

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