Set Theory/Constructing Numbers

Note: I have not checked this thoroughly.

One of the most familiar entities in mathematics are numbers. From pre-school to advanced levels, numbers are intimately linked with mathematics. But what are these numbers, mathematically speaking? This section will show the way mathematicians construct number systems. We will start from the axioms of set theory (ZFC Set Theory) and construct numbers in a purely set-theoritical method.

The Natural numbers $\mathbb {N}$

With these axioms in hand we can start building the numbers. Our focus here is to quick start and build all the commonly used numbers. Hence, we wont focus on developing all properties that sets have, trusting that the user will agree that our constructions are sound enough and proofs of all properties proved strictly through these axioms do exist.

Our first target is to make the basic counting numbers, also called natural numbers. The axiom of infinity says that a set X exists which satisfies the property that it contains the empty set and the set and, whenever a set y is a member of X, then $S(y)\!$ is also a member of X. There may or may not exist any other such sets. Taking the intersection of all such sets we get the set of natural numbers N. Clearly it is the smallest such set. Also as $\varnothing$ is a member of N, so is, $\{\varnothing \}$ , and so is $\{\varnothing \,\{\varnothing \}\}$ and so on. Intutively it is obvious that no other kind of members are there as N was the smallest such set. We let $0=\varnothing$ to match with our intuition that 0 should be defined as "nothing". Next we define $1=\{\varnothing \}$ to match with our intuition that 1 should convey the notion of "one-ness" which a set having a solitary element does. Similarly we define, $2=\{\varnothing \,\{\varnothing \}\}$ and so on.

Hence we have mathematically defined the set of natural numbers. We will define addition and multiplication for natural numbers as follows: Let $x^{+}=S(x)\!$ . Then we can recursively define addition by $a+(b^{+})=(a+b)^{+}$ . (Hence $1+1=1+0^{+}=(1+0)^{+}=1^{+}=2$ and so on. Note that $0^{+}=1$ .) Similarly multiplication is defined recursively by x×0 = 0 and x×S(y) = (x×y) + x. Such definitions are justified owing to a particular theorem in maths called recursion theorem, but we needn't go there. A few examples are usually sufficient to convince oneself of the validity of these definitions.

The Integers

Once more we will take a leap and assume that the reader trusts us when we say that all usual properties of natural numbers can be established in our constructed system of natural numbers. The reason we avoid doing that here is because we wish to avoid getting tied up in formal arguments, and sidetrack our goal to provide a quick mathematical build-up of the numbers.

Next we tackle the integers. Since we need some brief extraneous concepts in our toolbox for this purpose (which all are defined and their properties established consistent with the axioms of set theory) we shall briefly mention those. First up is the concept of an ordered pair (a,b) which is defined as the set {{a},{a,b}} and using this a 3-tuple (a,b,c) is defined as the ordered pair (a,(b,c)) i.e. {{a}{{b},{b,c}}}. Similary 4-tuples, 5-tuples etc may be defined. Next is the concept of functions and relations which we trust the reader is aware of from calculus. Finally, we talk of equivalence relations and equivalence classes.

A given relation ~ on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Equivalently, for all a, b and c in A:

a ~ a. (Reflexivity)
if a ~ b then b ~ a. (Symmetry)
if a ~ b and b ~ c then a ~ c. (Transitivity)

The equivalence class of a under ~, denoted [a], is defined as $[a]=\{b\in A|a\sim b\}$ . For example, if A is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars. Another example deals with functions: any function f : X → Y defines an equivalence relation on X by x1 ~ x2 if and only if f(x1) = f(x2). The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. the class [x] is the inverse image of f(x).

Red Points represent ordered pairs of natural numbers. Linked red points are equivalence classes representing the blue integers at the end of the line.

Now the set of integers can be formally defined as the set of equivalence classes of a particular relation. That relation is defined between ordered pairs of natural numbers (not between the natural numbers themselves but between their pairs).

The intuition is that (a, b) stands for the result of subtracting b from a. To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule:

(a,b)\sim (c,d)\,\!

precisely when

a+d=b+c.\,\!

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; denoting by [(a,b)] the equivalence class having (a,b) as a member, one has:

[(a,b)]+[(c,d)]:=[(a+c,b+d)].\,

[(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)].\,

The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:

-[(a,b)]:=[(b,a)].\,

Hence subtraction can be defined as the addition of the additive inverse:

[(a,b)]-[(c,d)]:=[(a+d,b+c)].\,

The standard ordering on the integers is given by:

[(a,b)]<[(c,d)]\,

if and only if

a+d<b+c.\,

It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.

Every equivalence class has a unique member that is of the form (n,0) or (0,n) (or both at once). The natural number n is identified with the class [(n,0)] (in other words the natural numbers are embedded into the integers by map sending n to [(n,0)]), and the class [(0,n)] is denoted −n (this covers all remaining classes, and gives the class [(0,0)] a second time since −0 = 0.

Thus, [(a,b)] is denoted by

{\begin{cases}a-b,&{\mbox{if }}a\geq b\\-(b-a),&{\mbox{if }}a<b.\end{cases}}

If the natural numbers are identified with the corresponding integers, this convention creates no ambiguity.

This notation recovers the familiar representation of the integers as {... −3,−2,−1, 0, 1, 2, 3, ...}.

Some examples are:

{\begin{aligned}0&=[(0,0)]&=[(1,1)]&=\cdots &&=[(k,k)]\\1&=[(1,0)]&=[(2,1)]&=\cdots &&=[(k+1,k)]\\-1&=[(0,1)]&=[(1,2)]&=\cdots &&=[(k,k+1)]\\2&=[(2,0)]&=[(3,1)]&=\cdots &&=[(k+2,k)]\\-2&=[(0,2)]&=[(1,3)]&=\cdots &&=[(k,k+2)].\end{aligned}}

The Rational numbers

Next stop: the rationals!

Those who have done a course in algebra may recognize the construction that follows as precisely being the one by which an integral domain is embedded in a field. (And indeed it is no big wonder as the integers form an integral domain and the rationals a field.) For those who think fields are open spaces and are thinking about agriculture, not to worry. We provide the full construction below.

a diagram showing a representation of the equivalent classes of pairs of integers

Mathematically the construction is similar as when we constructed the integers out of the naturals. We consider the following relation between all ordered pairs of integers (m,n), with n ≠ 0. Specifically, the relation is (m₁,n₁) ~ (m₂,n₂) if, and only if, m₁n₂ − m₂n₁ = 0. We can define addition and multiplication of these pairs with the following rules:

\left(m_{1},n_{1}\right)+\left(m_{2},n_{2}\right):=\left(m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}\right)

\left(m_{1},n_{1}\right)\times \left(m_{2},n_{2}\right):=\left(m_{1}m_{2},n_{1}n_{2}\right)

and, if m₂ ≠ 0, division by

{\frac {\left(m_{1},n_{1}\right)}{\left(m_{2},n_{2}\right)}}:=\left(m_{1}n_{2},n_{1}m_{2}\right).

The equivalence relation (m₁,n₁) ~ (m₂,n₂) if, and only if, m₁n₂ − m₂n₁ = 0 is a congruence relation, i.e. it is compatible with the addition and multiplication defined above, and we may define the set of all rational numbers Q to be the set of all equivalence classes under this relation. We denote by [(m₁,n₁)] the equivalence class containing (m₁,n₁). If (m₁,n₁) ~ (m₂,n₂) then, by definition, (m₁,n₁) belongs to [(m₂,n₂)] and (m₂,n₂) belongs to [(m₁,n₁)]; in this case we can write [(m₁,n₁)] = [(m₂,n₂)]. Given any equivalence class [(m,n)] there are a countably infinite number of representation, since

\cdots =[(-2m,-2n)]=[(-m,-n)]=[(m,n)]=[(2m,2n)]=\cdots .

The canonical choice for [(m,n)] is chosen so that gcd(m,n) = 1, i.e. m and n share no common factors, i.e. m and n are coprime. For example, we would write [(1,2)] instead of [(2,4)] or [(−12,−24)], even though [(1,2)] = [(2,4)] = [(−12,−24)].

The integers may be considered to be rational numbers by the embedding that maps m to [(m, 1)].

The Reals

The Real Numbers await us!

For starters we need the concept of Dedekind cuts. A Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element. The cut itself is, conceptually, the "gap" defined between A and B. In other words, A contains every rational number less than the cut, and B contains every rational number greater than the cut.

For convenience we may take the lower set A as the representative of any given Dedekind cut A, B, since A completely determines B. We now define the set of real numbers to be the set of all Dedekind cuts. In more detail, a real number $r$ is any subset of the set ${\textbf {Q}}$ of rational numbers that fulfills the following conditions:

$r$ is not empty
$r\neq {\textbf {Q}}$
r is closed downwards. In other words, for all $x,y\in {\textbf {Q}}$ such that $x<y$ , if $y\in r$ then $x\in r$
r contains no greatest element. In other words, there is no $x\in r$ such that for all $y\in r$ , $y\leq x$

We form the set ${\textbf {R}}$ of real numbers as the set of all Dedekind cuts $A$ of ${\textbf {Q}}$ .

We embed the rational numbers into the reals by identifying the rational number $q$ with the set of all smaller rational numbers $\{x\in {\textbf {Q}}:x<q\}$ .

Addition. $A+B:=\{a+b:a\in A\land b\in B\}$

Defining multiplication is less straightforward.
- if $A,B\geq 0$ then $A\times B:=\{a\times b:a\geq 0\land a\in A\land b\geq 0\land b\in B\}\cup \{x\in \mathrm {Q} :x<0\}$
- if either $A\,$ or $B\,$ is negative, we use the identities $A\times B=-(A\times -B)=-(-A\times B)=(-A\times -B)\,$ to convert $A\,$ and/or $B\,$ to positive numbers and then apply the definition above.

The Complex numbers

Are we done? Not quite. The set of complex numbers which the reader has no doubt encountered still awaits us. Fortunately, it does not require such a detailed development as the reals and we quickly obtain it as follows.

Let C be the set R² of ordered pairs (a, b) of real numbers. C is defined as the set of complex numbers. In this notation, the above formulas for addition and multiplication read