Arithmetization — Read

From $\mathbb{N}$source 11 to $\mathbb{Z}$source 11

Here are two basic realisations:

1. Every integer can be written in the form $n-m$source 15, with $n,m\in \mathbb{N}$source 15.

2. The information encoded in an expression $n-m$source 16 can equally be encoded by an ordered pair $⟨n,m⟩$source 16.

We already know that the ordered pairs of natural numbers are the elements of ${\mathbb{N}}^2$source 18. And we are assuming that we understand $\mathbb{N}$source 18. So here is a naïve suggestion, based on the two realisations we have had: let's treat integers as ordered pairs of natural numbers.

In fact, this suggestion is too naïve. Obviously we want it to be the case that $0-2=4-6$source 20. But evidently $⟨0,2⟩\ne ⟨4,6⟩$source 20. So we cannot simply say that ${\mathbb{N}}^2$source 20 is the set of integers.

Generalising from the preceding problem, what we want is the following: $$a-b=c-d\text{iff}a+d=c+b$$source 23 (It should be obvious that this is how integers are meant to behave: just add $b$source 24 and $d$source 24 to both sides.) And the easy way to guarantee this behaviour is just to define an equivalence relation between ordered pairs, $\sim$source 24, as follows: $$⟨a,b⟩\sim ⟨c,d⟩\text{iff}a+d=c+b$$source 25 We now have to show that this is an equivalence relation.

Now we can use this equivalence relation to take equivalence classes:

Now, one might have plenty of different philosophical reactions to this stipulative definition. Before we consider those reactions, though, it is worth continuing with some of the technicalities.

Having said what the integers are, we shall need to define basic functions and relations on them. Let's write ${\left[m,n\right]}_{\sim }$source 47 for the equivalence class under $\sim$source 47 with $⟨m,n⟩$source 47 as an element.Note: using the notation introduced in the definition of an equivalence class, we would have written ${\left[⟨m,n⟩\right]}_{\sim }$source 47 for the same thing. But that's just a bit harder to read. That is: $${\left[m,n\right]}_{\sim }=\{⟨a,b⟩\in {\mathbb{N}}^2:⟨a,b⟩\sim ⟨m,n⟩\}$$source 48 So now we offer some definitions: $$\begin{array}{cc}{\left[a,b\right]}_{\sim }+{\left[c,d\right]}_{\sim }& ={\left[a+c,b+d\right]}_{\sim }\\ {\left[a,b\right]}_{\sim }\times {\left[c,d\right]}_{\sim }& ={\left[ac+bd,ad+bc\right]}_{\sim }\\ {\left[a,b\right]}_{\sim }\le {\left[c,d\right]}_{\sim }& \text{iff}a+d\le b+c\end{array}$$source 50 (As is common, I'm using `$ab$source 55' to stand for `$(a\times b)$source 55', just to make the axioms easier to read.) Now, we need to make sure that these definitions behave as they ought to. Spelling out what this means, and checking it through, is rather laborious; we relegate the details to the Ordered Rings and Fields section. But the short point is: everything works!

One final thing remains. We have constructed the integers using natural numbers. But this will mean that the natural numbers are not themselves integers. We will return to the philosophical significance of this in the Some Philosophical Reflections section. On a purely technical front, though, we will need some way to be able to treat natural numbers as integers. The idea is quite easy: for each $n\in \mathbb{N}$source 62, we just stipulate that $n_{\mathbb{Z}}={\left[n,0\right]}_{\sim }$source 63. We need to confirm that this definition is well-behaved, i.e., that for any $m,n\in \mathbb{N}$source 64 $$\begin{array}{cc}{(m+n)}_{\mathbb{Z}}& =m_{\mathbb{Z}}+n_{\mathbb{Z}}\\ {(m\times n)}_{\mathbb{Z}}& =m_{\mathbb{Z}}\times n_{\mathbb{Z}}\\ m\le n& \leftrightarrow m_{\mathbb{Z}}\le n_{\mathbb{Z}}\end{array}$$source 66 But this is all pretty straightforward. For example, to show that the second of these obtains, we can simply help ourselves to the behaviour of the natural numbers and reason as follows:

$$\begin{array}{cc}{(m\times n)}_{\mathbb{Z}}& ={\left[m\times n,0\right]}_{\sim }\\ & ={\left[m\times n+0\times 0,m\times 0+0\times n\right]}_{\sim }\\ & ={\left[m,0\right]}_{\sim }\times {\left[n,0\right]}_{\sim }\\ & =m_{\mathbb{Z}}\times n_{\mathbb{Z}}\end{array}$$source 76

We leave it as an exercise to confirm that the other two conditions hold.

From $\mathbb{Z}$source 10 to $\mathbb{Q}$source 10

We just saw how to construct the integers from the natural numbers, using some naïve set theory. We shall now see how to construct the rationals from the integers in a very similar way. Our initial realisations are:

1. Every rational can be written in the form $\frac{i}{j}$source 17, where both $i$source 18 and $j$source 18 are integers but $j$source 18 is non-zero.

2. The information encoded in an expression $\frac{i}{j}$source 19 can equally be encoded in an ordered pair $⟨i,j⟩$source 20.

The obvious approach would be to think of the rationals as ordered pairs drawn from $\mathbb{Z}\times (\mathbb{Z}\setminus \{0_{\mathbb{Z}}\left\}\right)$source 23. As before, though, that would be a bit too naïve, since we want $\frac{3}{2}=\frac{6}{4}$source 25, but $⟨3,2⟩\ne ⟨6,4⟩$source 25. More generally, we will want the following: $$\frac{a}{b}=\frac{c}{d}\text{iff}a\times d=b\times c$$source 27 To get this, we define an equivalence relation on $\mathbb{Z}\times (\mathbb{Z}\setminus \{0_{\mathbb{Z}}\left\}\right)$source 30 thus: $$⟨a,b⟩\backsim ⟨c,d⟩\text{iff}a\times d=b\times c$$source 32 We must check that this is an equivalence relation. This is very much like the case of $\sim$source 36, and we will leave it as an exercise.

But it allows us to say:

As with the integers, we also want to define some basic operations. Where ${\left[i,j\right]}_{\backsim }$source 48 is the equivalence class under $\backsim$source 49 with $⟨i,j⟩$source 49 as an element, we say: $$\begin{array}{cc}{\left[a,b\right]}_{\backsim }+{\left[c,d\right]}_{\backsim }& ={\left[ad+bc,bd\right]}_{\backsim }\\ {\left[a,b\right]}_{\backsim }\times {\left[c,d\right]}_{\backsim }& ={\left[ac,bd\right]}_{\backsim }.\\ \text{To define}r\le s\text{on these rationals, we use the fact that}r\le s\text{iff}s-r\text{is not negative, i.e.,}r-s\text{can be written as}\frac{i}{j}\text{with}i\text{non-negative and}j\text{positive:}\hfill & {\left[a,b\right]}_{\backsim }\le {\left[c,d\right]}_{\backsim }& \text{iff}{\left[c,d\right]}_{\backsim }-{\left[a,b\right]}_{\backsim }={\left[i_{\mathbb{Z}},j_{\mathbb{Z}}\right]}_{\backsim }\end{array}$$source 50 for some $i\in \mathbb{N}$source 60 and $0\ne j\in \mathbb{N}$source 60.

We then need to check that these definitions behave as they ought to; and we relegate this to the Ordered Rings and Fields section. But they indeed do! Finally, we want some way to treat integers as rationals; so for each $i\in \mathbb{Z}$source 65, we stipulate that $i_{\mathbb{Q}}={\left[i,1_{\mathbb{Z}}\right]}_{\backsim }$source 65. Again, we check that all of this behaves correctly in the Ordered Rings and Fields section.

The Real Line

The next step is to show how to construct the reals from the rationals. Before that, we need to understand what is distinctive about the reals.

The reals behave very much like the rationals. (Technically, both are examples of ordered fields; for the definition of this, see the definition of an ordered field.) Now, if you worked through the exercises to the Size of Sets chapter, you will know that there are strictly more reals than rationals, i.e., that $\mathbb{Q}\prec \mathbb{R}$source 20. This was first proved by Cantor. But it's been known for about two and a half millennia that there are irrational numbers, i.e., reals which are not rational. Indeed:

In passing, this diagrammatic proof allows us to revisit the material from the More Myth than History section. Tennenbaum (1927–2006) was a thoroughly modern mathematician; but the proof is undeniably lovely, completely rigorous, and appeals to geometric intuition!

In any case: the reals are “more expansive” than the rationals. In some sense, there are “gaps” in the rationals, and these are filled by the reals. Weierstrass realised that this describes a single property of the real numbers, which distinguishes them from the rationals, namely the Completeness Property: Every non-empty set of real numbers with an upper bound has a least upper bound.

It is easy to see that the rationals do not have the Completeness Property. For example, consider the set of rationals less than $\sqrt{2}$source 76, i.e.: $$\{p\in \mathbb{Q}:p^2<2\text{or}p<0\}$$source 77 This has an upper bound in the rationals; its elements are all smaller than $3$source 80, for example. But what is its least upper bound? We want to say `$\sqrt{2}$source 80'; but we have just seen that $\sqrt{2}$source 80 is not rational. And there is no least rational number greater than $\sqrt{2}$source 80. So the set has an upper bound but no least upper bound. Hence the rationals lack the Completeness Property.

By contrast, the continuum “morally ought” to have the Completeness Property. We do not just want $\sqrt{2}$source 82 to be a real number; we want to fill all the “gaps” in the rational line. Indeed, we want the continuum itself to have no “gaps” in it. That is just what we will get via Completeness.

From $\mathbb{Q}$source 10 to $\mathbb{R}$source 10

In essence, the Completeness Property shows that any point $\alpha$source 12 of the real line divides that line into two halves perfectly: those for which $\alpha$source 14 is the least upper bound, and those for which $\alpha$source 14 is the greatest lower bound. To construct the real numbers from the rational numbers, Dedekind suggested that we simply think of the reals as the cuts that partition the rationals. That is, we identify $\sqrt{2}$source 18 with the cut which separates the rationals $<\sqrt{2}$source 19 from the rationals $>\sqrt{2}$source 19.

Let's tidy this up. If we cut the rational numbers into two halves, we can uniquely identify the partition we made just by considering its bottom half. So, getting precise, we offer the following definition:

So now we can say that $\sqrt{2}=\{p\in \mathbb{Q}:p^2<2\text{or}p<0\}$source 38. Of course, we need to check that this is a cut, but we relegate that to the Ordered Rings and Fields section.

As before, having defined some entities, we next need to define basic functions and relations upon them. We begin with an easy one: $$\alpha \le \beta \text{iff}\alpha \subseteq \beta$$source 44 This definition of an order allows to state the central result, that the set of cuts has the Completeness Property. Spelled out fully, the statement has this shape. If $S$source 49 is a non-empty set of cuts with an upper bound, then $S$source 50 has a least upper bound. In more detail: there is a cut, $\lambda$source 50, which is an upper bound for $S$source 50, i.e. $(\forall \alpha \in S)\alpha \subseteq \lambda$source 50, and $\lambda$source 51 is the least such cut, i.e. $(\forall \beta \in \mathbb{R})\left(\right(\forall \alpha \in S)\alpha \subseteq \beta \to \lambda \subseteq \beta )$source 51. Now here is the proof of the result:

So we have a bunch of entities which satisfy the Completeness Property. And one way to put this is: there are no “gaps” in our cuts. (So: taking further “cuts” of reals, rather than rationals, would yield no interesting new objects.)

Next, we must define some operations on the reals. We start by embedding the rationals into the reals by stipulating that $p_{\mathbb{R}}=\{q\in \mathbb{Q}:q<p\}$source 86 for each $p\in \mathbb{Q}$source 87. We then define: $$\begin{array}{cc}\alpha +\beta & =\{p+q:p\in \alpha \wedge q\in \beta \}\\ \alpha \times \beta & =\{p\times q:0\le p\in \alpha \wedge 0\le q\in \beta \}\cup 0^R& \text{if}\alpha ,\beta \ge 0_{\mathbb{R}}\end{array}$$source 88 To handle the other multiplication cases, first let: $$\begin{array}{cc}-\alpha & =\{p-q:p<0\wedge q\notin \alpha \}\end{array}$$source 97 and then stipulate: $$\begin{array}{cc}\alpha \times \beta & =\{\begin{array}{cc}-\alpha \times -\beta & \text{if}\alpha <0_{\mathbb{R}}\text{and}\beta <0_{\mathbb{R}}\\ -(-\alpha \times \beta )& \text{if}\alpha <0_{\mathbb{R}}\text{and}\beta >0_{\mathbb{R}}\\ -(\alpha \times -\beta )& \text{if}\alpha >0_{\mathbb{R}}\text{and}\beta <0_{\mathbb{R}}\end{array}\end{array}$$source 101 We then need to check that each of these definitions always yields a cut. And finally, we need to go through an easy (but long-winded) demonstration that the cuts, so defined, behave exactly as they should. But we relegate all of this to the Ordered Rings and Fields section.

Some Philosophical Reflections

So much for the technicalities. But what did they achieve?

Well, pretty uncontestably, they gave us some lovely pure mathematics. Moreover, there were some deep conceptual achievements. It was a profound insight, to see that the Completeness Property expresses the crucial difference between the reals and the rationals. Moreover, the explicit construction of reals, as Dedekind cuts, puts the subject matter of analysis on a firm footing. We know that the notion of a complete ordered field is coherent, for the cuts form just such a field.

For all that, we should air a few reservations about these achievements.

First, it is not clear that thinking of reals in terms of cuts is any more rigorous than thinking of reals in terms of their familiar (possibly infinite) decimal expansions. This latter “construction” of the reals has some resemblance to the construction of the reals via Cauchy sequence; but in fact, it was essentially known to mathematicians from the early 17th century onwards (see the appendix on the reals as Cauchy sequences). The real increase in rigour came from the realisation that the reals have the Completeness Property; the ability to construct real numbers as particular sets is perhaps not, by itself, so very interesting.

It is even less clear that the (much easier) arithmetization of the integers, or of the rationals, increases rigour in those areas. Here, it is worth making a simple observation. Having constructed the integers as equivalence classes of ordered pairs of naturals, and then constructed the rationals as equivalence classes of ordered pairs of integers, and then constructed the reals as sets of rationals, we immediately forget about the constructions. In particular: no one would ever want to invoke these constructions during a mathematical proof (excepting, of course, a proof that the constructions behaved as they were supposed to). It's much easier to speak about a real, directly, than to speak about some set of sets of sets of sets of sets of sets of sets of naturals.

It is most doubtful of all that these definitions tell us what the integers, rationals, or reals are, metaphysically speaking. That is, it is doubtful that the reals (say) are certain sets (of sets of sets … ). The main barrier to such a view is that the construction could have been done in many different ways. In the case of the reals, there are some genuinely interestingly different constructions (see the appendix on the reals as Cauchy sequences). But here is a really trivial way to obtain some different constructions: as in the Philosophical Reflections section on relations, we could have defined ordered pairs slightly differently; if we had used this alternative notion of an ordered pair, then our constructions would have worked precisely as well as they did, but we would have ended up with different objects. As such, there are many rival set-theoretic constructions of the integers, the rationals, and the reals. And now it would just be arbitrary (and embarrassing) to claim that the integers (say) are these sets, rather than those. (As in the Philosophical Reflections section on relations, this is an instance of an argument made famous by Paul Benacerraf 1965.)

A further point is worth raising: there is something quite odd about our constructions. We started with the natural numbers. We then construct the integers, and construct “the $0$source 83 of the integers”, i.e., ${\left[0,0\right]}_{\sim }$source 84. But $0\ne {\left[0,0\right]}_{\sim }$source 84. Indeed, given our constructions, no natural number is an integer. But that seems extremely counter-intuitive. Indeed, in the Some Important Sets section, we claimed without much argument that $\mathbb{N}\subseteq \mathbb{Q}$source 88. If the constructions tell us exactly what the numbers are, this claim was trivially false.

Standing back, then, where do we get to? Working in a naïve set theory, and helping ourselves to the naturals, we are able to treat integers, rationals, and reals as certain sets. In that sense, we can embed the theories of these entities within a set theory. But the philosophical import of this embedding is just not that straightforward.

Of course, none of this is the last word! The point is only this. Showing that the arithmetization of the reals is of deep philosophical significance would require some additional philosophical argument.

Ordered Rings and Fields

Throughout this chapter, we claimed that certain definitions behave “as they ought”. In this technical appendix, we will spell out what we mean, and (sketch how to) show that the definitions do behave “correctly”.

In the From the Natural Numbers to the Integers section, we defined addition and multiplication on $\mathbb{Z}$source 18. We want to show that, as defined, they endow $\mathbb{Z}$source 19 with the structure we “would want” it to have. In particular, the structure in question is that of a commutative ring.

So, to check that the integers form a commutative ring, we just need to check that we meet these eight conditions. None of the conditions is difficult to establish, but this is a bit laborious. For example, here is how to prove Associativity, in the case of addition:

Equally, here is how to prove Additive Inverse:

And here is a proof of Distributivity:

We leave it as an exercise to prove the remaining five conditions. Having done that, we have shown that $\mathbb{Z}$source 93 constitutes a commutative ring, i.e., that addition and multiplication (as defined) behave as they should.

But our task is not over. As well as defining addition and multiplication over $\mathbb{Z}$source 102, we defined an ordering relation, $\le$source 102, and we must check that this behaves as it should. In more detail, we must show that $\mathbb{Z}$source 104 constitutes an ordered ring.Recall from the definition of a linear order that a total order is a relation which is reflexive, transitive, anti-symmetric, and connected. In the context of order relations, connectedness is sometimes called trichotomy, since for any $a$source 108 and $b$source 108 we have $a\le b\vee a=b\vee a\ge b$source 108.

As before, it is laborious but routine to show that $\mathbb{Z}$source 124, as constructed, is an ordered ring. We will leave that to you.

This takes care of the integers. But now we need to show very similar things of the rationals. In particular, we now need to show that the rationals form an ordered field, under our given definitions of $+$source 130, $\times$source 130, and $\le$source 130:

Once you have shown that $\mathbb{Z}$source 138 constitutes an ordered ring, it is easy but laborious to show that $\mathbb{Q}$source 139 constitutes an ordered field.

Having dealt with the integers and the rationals, it only remains to deal with the reals. In particular, we need to show that $\mathbb{R}$source 146 constitutes a complete ordered field, i.e., an ordered field with the Completeness Property. Now, the theorem that the set of cuts has the Completeness Property established that $\mathbb{R}$source 149 has the Completeness Property. However, it remains to run through the (tedious) details of checking that $\mathbb{R}$source 150 is an ordered field.

Before tearing off into that laborious exercise, we need to check some more “immediate” things. For example, we need a guarantee that $\alpha +\beta$source 155, as defined, is indeed a cut, for any cuts $\alpha$source 156 and $\beta$source 156. Here is a proof of that fact:

Similar efforts will allow you to check that $\alpha -\beta$source 170 and $\alpha \times \beta$source 171 and $\alpha ÷\beta$source 171 are cuts (in the last case, ignoring the case where $\beta$source 172 is the zero-cut). Again, though, we will simply leave this to you.

But here is a small loose end to tidy up. In the From the Rationals to the Reals section, we suggest that we can take $\sqrt{2}=\{p\in \mathbb{Q}:p<0\text{or}{p}^2<2\}$source 180. But we do need to show that this set is a cut. Here is a proof of that fact:

Appendix: the Reals as Cauchy Sequences

In the From the Rationals to the Reals section, we constructed the reals as Dedekind cuts. In this section, we explain an alternative construction. It builds on Cauchy's definition of (what we now call) a Cauchy sequence; but the use of this definition to construct the reals is due to other nineteenth-century authors, notably Weierstrass, Heine, Méray and Cantor. (For a nice history, see John J. O'Connor and Edmund F. Robertson 2005.)

Before we get to the nineteenth century, it's worth considering Simon Stevin (1548–1620). In brief, Stevin realised that we can think of each real in terms of its decimal expansion. Thus even an irrational number, like $\sqrt{2}$source 24, has a nice decimal expansion, beginning: $$1.41421356237\dots$$source 25 It is very easy to model decimal expansions in set theory: simply consider them as functions $d:\mathbb{N}\to \mathbb{N}$source 29, where $d\left(n\right)$source 29 is the $n$source 30-th decimal place that we are interested in. We will then need a bit of tweak, to handle the bit of the real number that comes before the decimal point (here, just $1$source 32). We will also need a further tweak (an equivalence relation) to guarantee that, for example, $0.999\dots =1$source 33. But it is not difficult to offer a perfectly rigorous construction of the real numbers, in the manner of Stevin, within set theory.

Stevin is not our focus. (For more on Stevin, see Karin Usadi Katz and Mikhail G. Katz 2012.) But here is a closely related thought. Instead of treating $\sqrt{2}$source 40's decimal expansion directly, we can instead consider a sequence of increasingly accurate rational approximations to $\sqrt{2}$source 42, by considering the increasingly precise expansions: $$1,1.4,1.414,1.4142,1.41421,\dots$$source 44 The idea that reals can be considered via “increasingly good approximations” provides us with the basis for another sequence of insights (akin to the realisations that we used when constructing $\mathbb{Q}$source 50 from $\mathbb{Z}$source 50, or $\mathbb{Z}$source 50 from $\mathbb{N}$source 50). The basic insights are these:

1. Every real can be written as a (perhaps infinite) decimal expansion.

2. The information encoded by a (perhaps infinite) decimal expansion can be equally be encoded by a sequence of rational numbers.

3. A sequence of rational numbers can be thought of as a function from $\mathbb{N}$source 59 to $\mathbb{Q}$source 59; just let $f\left(n\right)$source 59 be the $n$source 59-th rational in the sequence.

Of course, not just any function from $\mathbb{N}$source 62 to $\mathbb{Q}$source 62 will give us a real number. For instance, consider this function: $$f\left(n\right)=\{\begin{array}{cc}1& \text{if}n\text{is odd}\\ 0& \text{if}n\text{is even}\end{array}$$source 64 Essentially the worry here is that the sequence $0,1,0,1,0,1,0,\dots$source 70 doesn't seem to “hone in” on any real. So: to ensure that we consider sequences which do hone in on some real, we need to restrict our attention to sequences which have some limit.

We have already encountered the idea of a limit, in the Rigorous Definition of Limits section. But we cannot use quite the same definition as we used there. The expression “$(\forall \epsilon >0)$source 77” there tacitly involved quantification over the real numbers; and we were considering the limits of functions on the real numbers; so invoking that definition would be to help ourselves to the real numbers; and they are exactly what we were aiming to construct. Fortunately, we can work with a closely related idea of a limit.

The general idea of a limit is the same as before: if you want a certain level of precision (measured by $\epsilon$source 90), there is a “region” to look in (any input greater than $\ell$source 91). And it is easy to see that our sequence $1$source 92, $1.4$source 92, $1.414$source 92, $1.4142$source 92, $1.41421$source 93 … has a limit: if you want to approximate $\sqrt{2}$source 93 to within an error of $\frac{1}{{10}^n}$source 94, then just look to any entry after the $n$source 95-th.

The obvious thought, then, would be to say that a real number just is any Cauchy sequence. But, as in the constructions of $\mathbb{Z}$source 98 and $\mathbb{Q}$source 99, this would be too naïve: for any given real number, multiple different Cauchy sequences indicate that real number. A simple way to see this is as follows. Given a Cauchy sequence $f$source 101, define $g$source 102 to be exactly the same function as $f$source 102, except that $g\left(0\right)\ne f\left(0\right)$source 102. Since the two sequences agree everywhere after the first number, we will (ultimately) want to say that they have the same limit, in the sense employed in the definition of a Cauchy sequence, and so should be thought of “defining” the same real. So, we should really think of these Cauchy sequences as the same real number.

Consequently, we again need to define an equivalence relation on the Cauchy sequences, and identify real numbers with equivalence classes. First we need the idea of a function which tends to $0$source 111 in the limit. For any function $h:\mathbb{N}\to \mathbb{Q}$source 112, say that $h$source 112 tends to $0$source 113 iff for any positive $\epsilon \in \mathbb{Q}$source 113 we have that $(\exists \ell \in \mathbb{N})(\forall n>\ell )\left|h\right(n\left)\right|<\epsilon$source 114.Compare this with the definition of $\underset{x\to \infty }{\lim }f\left(x\right)=0$source 115 in the Rigorous Definition of Limits section. Further, where $f$source 117 and $g$source 117 are functions $\mathbb{N}\to \mathbb{Q}$source 118, let $(f-g)\left(n\right)=f\left(n\right)-g\left(n\right)$source 118. Now define: $$f\Bumpeq g\text{iff}(f-g)\text{tends to}0.$$source 119 We need to check that $\Bumpeq$source 122 is an equivalence relation; and it is. We can then, if we like, define the reals as the equivalence classes, under $\Bumpeq$source 124, of all Cauchy sequences from $\mathbb{N}\to \mathbb{Q}$source 124.

Having done this, we shall as usual write ${\left[f\right]}_{\Bumpeq }$source 133 for the equivalence class with $f$source 134 as an element. However, to keep things readable, in what follows we will drop the subscript and write just $\left[f\right]$source 136. We also stipulate that, for each $q\in \mathbb{Q}$source 136, we have $q_{\mathbb{R}}=\left[c_q\right]$source 137, where $c_q$source 137 is the constant function $c_q\left(n\right)=q$source 138 for all $n\in \mathbb{N}$source 138. We then define basic relations and operations on the reals, e.g.: $$\begin{array}{cc}\left[f\right]+\left[g\right]& =\left[(f+g)\right]\\ \left[f\right]\times \left[g\right]& =\left[(f\times g)\right]\end{array}$$source 140 where $(f+g)\left(n\right)=f\left(n\right)+g\left(n\right)$source 144 and $(f\times g)\left(n\right)=f\left(n\right)\times g\left(n\right)$source 144. Of course, we also need to check that each of $(f+g)$source 145, $(f-g)$source 146 and $(f\times g)$source 146 are Cauchy sequences when $f$source 146 and $g$source 146 are; but they are, and we leave this to you.

Finally, we define a notion of order. Say $\left[f\right]$source 149 is positive iff both $\left[f\right]\ne 0_{\mathbb{R}}$source 150 and $(\exists \ell \in \mathbb{N})(\forall n>\ell )0<f\left(n\right)$source 150. Then say $\left[f\right]<\left[g\right]$source 151 iff $\left[(g-f)\right]$source 152 is positive. We have to check that this is well-defined (i.e., that it does not depend upon choice of “representative” function from the equivalence class).

But having done this, it is quite easy to show that these yield the right algebraic properties; that is:

It is harder to prove that the reals, so constructed, have the Completeness Property, so we will give the proof.