Dedekind cuts

The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system.Dedekind’s motivation behind this project is to notice that a real number $\alpha$, intuitively,is completely determined by the rationals strictly smaller than $\alpha$ and those strictly larger than$\alpha$. Concerning the completeness or continuity of the real line, Dedekind notes in [2] that

If all points of the straight line fall into two classes such that every point of the first class lies to theleft of every point of the second class, then there exists one and only one point which produces this divisionof all points into two classes, this severing of the straight line into two portions.

Dedekind defines a point to produce the division of the real line if this point is either theleast or greatest element of either one of the classes mentioned above. He further notes that the completenessproperty, as he just phrased it, is deficient in the rationals, which motivates the definition of reals as cutsof rationals. Because all rationals greater than $\alpha$ are reallyjust excess baggage, we prefer to swaysomewhat from Dedekind’s original definition. Instead, we adopt the following definition.

Definition.

A Dedekind cut is a subset $\alpha$ of the rational numbers $\mathbb{Q}$ that satisfies theseproperties:

1. $\alpha$ is not empty.

2. $\mathbb{Q}\setminus \alpha$ is not empty.

3. $\alpha$ contains no greatest element

4. For $x,y\in \mathbb{Q}$, if $x\in \alpha$ and , then $y\in \alpha$ as well.

Dedekind cuts are particularly appealing for two reasons. First, they make it very easy toprove the completeness, or continuity of the real line. Also, they make it quite plain todistinguish the rationals from the irrationals on the real line, and put the latter on a firmlogical foundation.In the construction of the real numbers from Dedekind cuts, we make the following definition:

Definition.

A real number is a Dedekind cut. We denote the set of all real numbers by $ℝ$ and weorder them by set-theoretic inclusion, that is to say, for any $\alpha ,\beta \in ℝ$,

where the inclusion is strict. We further define $\alpha =\beta$ as real numbersif $\alpha$ and $\beta$ are equal as sets. As usual, we write $\alpha \le \beta$if or $\alpha =\beta$. Moreover, a real number $\alpha$ is said to be irrationalif $\mathbb{Q}\setminus \alpha$ contains no least element.

The Dedekind completeness property of real numbers, expressed as the supremum property, now becomes straightforwardto prove. In what follows, we will reserve Greek variables for real numbers, and Roman variables for rationals.

Theorem 1.

Every nonempty subset of real numbers that is bounded above has a least upper bound.

Proof.

Let $A$ be a nonempty set of real numbers, such that for every $\alpha \in A$ we have that$\alpha \le \gamma$ for some real number $\gamma$. Now define the set

$$supA=\bigcup _{\alpha \in A}\alpha .$$

We must show that this set is a real number. This amounts to checking the four conditionsof a Dedekind cut.

1. $supA$ is clearly not empty, for it is the nonempty union of nonempty sets.

2. Because $\gamma$ is a real number, there is some rational $x$ that is not in $\gamma$. Sinceevery $\alpha \in A$ is a subset of $\gamma$, $x$ is not in any $\alpha$, so $x\notin supA$either. Thus, $\mathbb{Q}\setminus supA$ is nonempty.

3. If $supA$ had a greatest element $g$, then $g\in \alpha$for some $\alpha \in A$. Then $g$ would be a greatest element of $\alpha$, but $\alpha$ is a real number, soby contrapositive, $supA$ has no greatest element.

4. Lastly, if $x\in supA$, then $x\in \alpha$ for some $\alpha$, so given any because $\alpha$is a real number $y\in \alpha$, whence $y\in supA$.

Thus, $supA$ is a real number. Trivially, $supA$ is an upper bound of $A$, for every $\alpha \subseteq supA$.It now suffices to prove that $supA\le \gamma$, because $\gamma$ was an arbitraryupper bound. But this is easy, becauseevery $x\in supA$ is an element of $\alpha$ for some $\alpha \in A$, so because $\alpha \subseteq \gamma$,$x\in \gamma$. Thus,$supA$ is the least upper bound of $A$. We call this real number the supremum of A.∎

To finish the construction of the real numbers, we must endow them with algebraic operations, define the additiveand multiplicative identity elements, prove that these definitions give a field, and prove further results about the order of the reals (such as the totality of thisorder) – in short, build a complete ordered field. This task is somewhat laborious, but we include here theappropriate definitions. Verifying their correctness can be an instructive, albeit tiresome, exercise. We use thesame symbols for the operations on the reals as for the rational numbers; this should cause no confusion in context.

Definition.

Given two real numbers $\alpha$ and $\beta$, we define

• The additive identity, denoted $0$, is

• The multiplicative identity, denoted $1$, is

• Addition of $\alpha$ and $\beta$ denoted $\alpha +\beta$ is

  $$\alpha +\beta :=\{x+y:x\in \alpha ,y\in \beta \}$$

• The opposite of $\alpha$, denoted $-\alpha$, is

  $$-\alpha :=\{x\in \mathbb{Q}:-x\notin \alpha ,\text{ but }-x\text{ is not the least element of }\mathbb{Q}\setminus \alpha \}$$

• The absolute value of $\alpha$, denoted $|\alpha |$, is

• If $\alpha ,\beta >0$, then multiplication of $\alpha$ and $\beta$, denoted $\alpha \cdot \beta$, is

  $$\alpha \cdot \beta :=\{z\in \mathbb{Q}:z\le 0\text{ or }z=xy\text{ for some }x\in \alpha ,y\in \beta \text{ with }x,y>0\}$$

  In general,

• The inverse of $\alpha >0$, denoted ${\alpha }^{-1}$, is

  $${\alpha }^{-1}:=\{x\in \mathbb{Q}:x\le 0\text{ or }x>0\text{ and }(1/x)\notin \alpha ,\text{ but }1/x\text{ is not the least element of }\mathbb{Q}\setminus \alpha \}$$

  If ,

  $${\alpha }^{-1}:=-{(|\alpha |)}^{-1}$$

All that remains (!) is to check that the above definitions do indeed define a complete ordered field, and thatall the sets implied to be real numbers are indeed so.The properties of $ℝ$ as an ordered field follow from these definitions and the properties of $\mathbb{Q}$ as anordered field.It is important to point out that in two steps, in showing that inverses and opposites are properly defined,we require an extra property of $\mathbb{Q}$, not merely in its capacity as an ordered field.This requirement is the Archimedean property.

Moreover, because $ℝ$ is a field of characteristic $0$, it contains an isomorphic copy of $\mathbb{Q}$. The rationalscorrespond to the Dedekind cuts $\alpha$ for which $\mathbb{Q}\setminus \alpha$ contains a least member.