Confused about incompleteness of rational numbers

Question

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?

Answer 1

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.

Answer 2

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.