Why is $\{p \in \mathbb{Q},:p<1\} | \{p \in \mathbb{Q}: p \geq 1\}$ a real number?

Question

A basic example of a Dedekind cut is:

$A|B$ = $\{p \in \mathbb{Q}:p<1\} | \{p \in \mathbb{Q}: p \geq 1\}$

This is very confusing because what we have here is a pair of subsets of $\mathbb{Q}$ whereas (without knowing Dedekind cut) we are used to real number as being a singleton

For example, $1$ is a real number. We can think of $1$ as a singleton all by itself i.e $\{1\} \in \mathbb{R}$

• Is there any reconciliation between between singleton in $\mathbb{R}$ point of view and a Dedekind cut which is a pair of subsets of $\mathbb{Q}$ in defining a real number?

• Since $\{p \in \mathbb{Q}: p<1\} | \{p \in \mathbb{Q}: p \geq 1\}$ is a real number, which real number does it represent?

• Since $\{p \in \mathbb{Q}: p<c\} | \{p \in \mathbb{Q}: p \geq c\}$ is a real number, for some rational number $c \in \mathbb{Q}$, then why isn't K-12 taught using addition and multiplication of $\{p \in \mathbb{Q}: p<c\} | \{p \in \mathbb{Q}: p \geq c\}$ instead of $a,b \in \mathbb{R}$, since they are equivalent

Note: first time seeing Dedekind cut as ways to represent real numbers

Answer 1

We can identify a rational number $c \in \mathbb{Q}$ with its Dedekind cut

$$\{x \in \mathbb{Q}, \,x < c\}|\{x \in \mathbb{Q}, \,x \ge c\}$$

Of course, it might seem we've just introduced a fancy way to obfuscate the really simple notion of rational numbers. But the point of Dedekind is that there are Dedekind cuts that are not associated with a rational number, such as

$$\{x \in \mathbb{Q}, \,x^2 < 2\}|\{x \in \mathbb{Q}, \,x^2 \ge 2\}$$

This cut cannot correspond to any rational number $c$ because this number should satisfy $c^2 = 2$. So Dedekind cuts extend rational numbers. We can also see define operations (comparison, addition, multiplication, inversion) that extend those of rational numbers. So we can think of those cuts as lying on a number line, and we can prove this line has no "holes".

In summary, Dedekind cuts are a construction of the real numbers : this a set that has all the properties of the real numbers, namely :

• we can add, subtract, multiply, divide real numbers (more precisely $\mathbb{R}$ is a field)

• numbers lie on a line without holes (which means there's a total ordering, compatible with sums and products, and each set that has a majorant has a supremum)

• all these operations ($+,-,\times,/, \le$) extend those on $\mathbb{Q}$

The goal of this construction is to prove that such a set (real numbers and its properties) exists. Once the proof is done, you're free to get back thinking about them as numbers on a line.

Answer 2

The whole point of Dedekind cuts is to prove that there exists a complete, ordered field with $\mathbb{Q}$ (isomorphic to) a subfield. Once we have that field, we rename it $\mathbb{R}$ and identify* each of the cuts with a real number. So if you prefer, you can think of the elements of $\mathbb{R}$ as a collection of sets, but this is rarely useful.

As far as your second question, the cut whose lower set is $\{q \in \mathbb{Q} : q < 1\}$ gets identified with its supremum, $1$.

As far as why we don't teach this in K-12, just think about why we don't teach children addition from the perspective of ordinals and successor functions. Without a degree of mathematical maturity, it's too abstract to be useful. Or for a different analogy, why don't we teach children to read with Shakespeare?

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*identify == bijection