Defining real numbers by only one half of Dedekind Cuts

Question

Spivak defines a real number as a set $\beta$ of rational numbers with the following four properties:

(1) if $x$ is in $\beta$ and $y$ is a rational number with $y < x$, then $y$ is also in $\beta$
(2) $\beta$ is a nonempty set
(3) $\beta$ does not equal $\mathbb{Q}$ (set of rational numbers)
(4) there is no greatest element in $\beta$

Example: $2^{1/2} = \{ x : x < 0 \text{ or } x^2 < 2 \}$

My question is: why does Spivak define real numbers only by the lower side of a Dedekind cut? For reference, compare this definition to:

A cut in $\mathbb{Q}$ is a pair of subsets $A$, $B$ of $\mathbb{Q}$ such that

(a) $A∪B=\mathbb{Q}$, $A\ne\emptyset$, $B\ne\emptyset$, $A∩B=\emptyset$.

(b) if a is an element of A and b an element of B, then a is greater than b

(c) A contains no largest element

Example: (i) $1 = \{r∈\mathbb{Q}:r<1\}\cup\{r∈\mathbb{Q}:r≥1\}$.

Side question: in the above example, what exactly does the $x<0$ condition do? It seems to me superfluous.

Answer 1

It's a matter of convention. He could as well define a real number as the per $(\beta,\mathbb{Q}\setminus\beta)$. Or by $\mathbb{Q}\setminus\beta$.

Concerning the side question, it is not superfluous. If he hadn't added that condition, then we woulf have $0\in\beta$ and $-2\notin\beta$. That's impossible, by the definition of Dedekind cut.

Answer 2

The idea behind Dedekind cuts is to associate to every real number a unique partition of $\mathbb{Q}$ into a pair of subsets.

For an irrational number $r$, it's easy: consider $$ L_r=\{x\in\mathbb{Q}:x\le r\} \qquad U_r=\{x\in\mathbb{Q}:x\ge r\} $$ For rational $r$ there is a hindrance: what subset should $r$ belong to?

Leaving aside this problem for a moment, we see that the subsets $L_r$ and $U_r$ (for irrational $r$ satisfy the properties

1. $L_r\ne\emptyset$, $U_r\ne\emptyset$
2. $L_r\cap U_r=\emptyset$
3. $L_r\cup U_r=\mathbb{Q}$
4. if $x\in L_r$ and $y<x$, then $y\in L_r$
5. if $x\in U_r$ and $y>x$, then $y\in U_r$

The same would happen for rational $r$, with the exception of property 2. In order to deal with this case we need to decide what class $r$ should belong to. There are at least three ways to do it:

1. add the condition that $L_r$ has no maximum
2. add the condition that $U_r$ has no minimum
3. add the condition that $L_r\cap U_r$ contains at most one element

So we have three possible competing definitions of a Dedekind cut as a pair of subsets $A$ and $B$ of $\mathbb{Q}$ such that

Definition 1

1. $A\ne\emptyset$, $B\ne\emptyset$
2. $A\cap B=\emptyset$
3. $A\cup B=\mathbb{Q}$
4. if $x\in A$ and $y<x$, then $y\in A$
5. if $x\in B$ and $y>x$, then $y\in B$
6. $A$ has no maximum

Definition 2

1. $A\ne\emptyset$, $B\ne\emptyset$
2. $A\cap B=\emptyset$
3. $A\cup B=\mathbb{Q}$
4. if $x\in A$ and $y<x$, then $y\in A$
5. if $x\in B$ and $y>x$, then $y\in B$
6. $B$ has no minimum

Definition 3

1. $A\ne\emptyset$, $B\ne\emptyset$
2. $A\cap B$ contains at most one element
3. $A\cup B=\mathbb{Q}$
4. if $x\in A$ and $y<x$, then $y\in A$
5. if $x\in B$ and $y>x$, then $y\in B$

Choosing one among these is just a matter of personal preferences.

Spivak chose Definition 1, discarding the “upper set”, which can obviously be inferred from the “lower set”, given properties 2 and 3.

Answer 3

If we call something that satisfies Spivak's conditions a Spivak cut, then each Spivak cut defines a Dedekind cut, and vice versa. Once you're given $\beta$, it's trivial to define $B$ as the set of rationals that are in $\beta$, and $A$ as the set of rationals that are not. Presumably, Dedekind defined cuts as partitions because he wanted to work with both sets, while Spivak can do everything they want to do with the lower set.