4.1 Total orders and ordered fields

Why order needs its own note

Earlier chapters focused on building $N$ , $Z$ , and $Q$ , then defining their operations carefully. Chapter 4 changes the viewpoint. We now ask not only what the numbers are, but also what sort of order structure they carry.

That shift matters because later concepts such as bounds, supremum, limits, and completeness all depend on the order relation. Before talking about “least upper bounds,” we need to know what kind of order we are even using.

Partial order versus total order

Definition

Total order

A set $X$ with relation $\le$ is totally ordered if:

$x\le x$ for every $x\in X$ ,
$x\le y$ and $y\le x$ imply $x=y$ ,
$x\le y$ and $y\le z$ imply $x\le z$ ,
for every $x,y\in X$ , either $x\le y$ or $y\le x$ .

The first three conditions say that $\le$ is a partial order. The fourth condition is the extra comparability condition that makes the order total.

The crucial new feature is comparability. In a total order, there are no two elements left floating without comparison.

Worked example

A standard total order

The usual order on {1,2,3,4} is total. For any two elements, one lies at or to the left of the other, so comparison is never ambiguous.

Worked example

A partial order that is not total

Let $X={1,2,3,6}$ and declare $x\le y$ when x divides y.

This is a partial order, but not a total order. The elements 2 and 3 are incomparable: 2 does not divide 3, and 3 does not divide 2.

This example is important because it shows that “ordered” does not automatically mean “every pair is comparable.” That extra property has to be checked.

Restricted order on a subset

Once an ambient set is totally ordered, every subset inherits that order.

Theorem

Restricted order stays total

If $(X,\le)$ is totally ordered and $Y\subseteq X$ , then $Y$ becomes totally ordered when we use the same comparison rule on elements of $Y$ .

Why does this matter? Because later we will study subsets such as

S=\{q\in Q\mid q^2<2\}

inside a larger ordered set. We do not invent a new order on $S$ ; we restrict the one already living on $Q$ .

The standard order on $Z$ and $Q$

For integers and rationals, the familiar order can be written in a way that fits the algebra already built in earlier chapters:

x\le y \quad \text{iff} \quad 0\le y-x.

So comparison is translated into a statement about the sign of a difference. This is useful because sign, addition, and multiplication are already part of the algebraic language of $Z$ and $Q$ .

Theorem

The standard order on $Z$ and $Q$ is total

With the usual notion of positivity, the standard order on $Z$ and on $Q$ is a total order.

The reason is structural: for any difference $y-x$ , exactly one of three things happens. It is positive, it is zero, or it is negative. Those three cases give $x<y$ , $x=y$ , or $x>y$ , so every pair is comparable.

Order and operations must cooperate

The order is not an isolated decoration. It has to behave properly with the field operations.

Definition

Order-compatible algebra facts

For rational numbers x,y,z:

if $x\le y$ , then $x+z\le y+z$ ;
if $x\ge 0$ and $y\ge 0$ , then $xy\ge 0$ .

The first rule says translation preserves order. The second says multiplying nonnegative quantities cannot suddenly create a negative result.

Worked example

Why translation invariance matters

If 1/3\le 1/2, then adding 2 to both sides gives

2+\frac13 \le 2+\frac12.

This is not a new theorem each time. It is the same structural rule applied to different numbers.

Without these compatibility rules, the order would not interact correctly with the algebra. Later arguments about intervals, bounds, and absolute values would fall apart.

Fields and ordered fields

One can summarize almost everything we know about $Q$ by packaging the operations together.

Definition

Field

A field is a set with distinguished elements 0 and 1, together with addition and multiplication, such that:

addition and multiplication are associative and commutative,
multiplication distributes over addition,
every element has an additive inverse,
every nonzero element has a multiplicative inverse.

This definition remembers the algebra, but it says nothing about order.

Definition

Ordered field

An ordered field is a field equipped with a total order $\le$ such that:

$x\le y$ implies $x+z\le y+z$ ,
$x\ge 0$ and $y\ge 0$ imply $xy\ge 0$ .

So an ordered field is not just “a field plus a comparison symbol.” It is a field whose algebra and order fit together coherently.

The rational numbers $Q$ are an ordered field. The real numbers $R$ will also be an ordered field, but chapter 4 will show that $R$ has one more crucial property beyond this: completeness.

Common mistake

A total order is more than a left-to-right picture

Students often think a total order is just “something that can be drawn on a line.” The real point is the comparability axiom. A partial order can still have a clear diagram or hierarchy and yet fail to compare some pairs.

Common mistake

A field is not automatically an ordered field

The field axioms only describe algebraic operations. To become an ordered field, the set also needs a total order, and that order must respect addition and multiplication in the precise way stated above.

Quick checks

Quick check

Why is divisibility on positive integers not a total order?

Look for two incomparable elements.

Solution

Answer

Quick check

If $x\le y$ , why does $x-z\le y-z$ also hold?

Rewrite subtraction as addition.

Solution

Answer

Exercises

Quick check

Explain why every subset of a totally ordered set inherits a total order.

Take two elements of the subset and compare them in the ambient set.

Solution

Guided solution

Quick check

Why is $Q_{>0}$ not a field?

Check closure and inverses against the field axioms.

Solution

Guided solution

Read this after 3.4 Rationals and well-defined operations and 3.5 Gaps in Q and sqrt(2). Then continue to 4.2 Upper bounds, supremum, and infimum.

4.1 Total orders and ordered fields

MATH1090: Set theory

Why order needs its own note

Partial order versus total order

Total order

A standard total order

A partial order that is not total

Restricted order on a subset

Restricted order stays total

The standard order on $Z$ and $Q$

The standard order on $Z$ and $Q$ is total

Order and operations must cooperate

Order-compatible algebra facts

Why translation invariance matters

Fields and ordered fields

Field

Ordered field

A total order is more than a left-to-right picture

A field is not automatically an ordered field

Quick checks

Why is divisibility on positive integers not a total order?

Answer

If $x\le y$ , why does $x-z\le y-z$ also hold?

Answer

Exercises

Explain why every subset of a totally ordered set inherits a total order.

Guided solution

Why is $Q_{>0}$ not a field?

Guided solution

Prerequisites

Key terms in this unit

More notes in this series

4.1 Total orders and ordered fields

MATH1090: Set theory

Why order needs its own note

Partial order versus total order

Total order

A standard total order

A partial order that is not total

Restricted order on a subset

Restricted order stays total

The standard order on ZZZ and QQQ

The standard order on ZZZ and QQQ is total

Order and operations must cooperate

Order-compatible algebra facts

Why translation invariance matters

Fields and ordered fields

Field

Ordered field

A total order is more than a left-to-right picture

A field is not automatically an ordered field

Quick checks

Why is divisibility on positive integers not a total order?

Answer

If x≤yx\le yx≤y, why does x−z≤y−zx-z\le y-zx−z≤y−z also hold?

Answer

Exercises

Explain why every subset of a totally ordered set inherits a total order.

Guided solution

Why is Q>0Q_{>0}Q>0​ not a field?

Guided solution

Related notes

Prerequisites

Key terms in this unit

More notes in this series

The standard order on $Z$ and $Q$

The standard order on $Z$ and $Q$ is total

If $x\le y$ , why does $x-z\le y-z$ also hold?

Why is $Q_{>0}$ not a field?