Real numbers definition problem

Question

In the definition of the limit, real positive numbers are used as distance, so we can not use the limit in the definition of real numbers. the definition of the limit. But irrational numbers are defined as the limit of the sequence of rational numbers. Is not this a logical problem?

Answer 1

It is kind of a problem. Not in the sense that we don't know how to fix it, but because the fix is so relatively uninteresting that it doesn't really feel worth the effort to write it down.

You can develop much of the theory of metric spaces and limits by considering distances that are positive rationals only. This will be enough to allow you to define Cauchy sequences of rationals and make the push forward towards the real numbers.

Afterwards you'll then want to do everything over with real distances, for full generality. In particular you will then prove that it makes no difference whether the $\varepsilon$s and $\delta$s in the definitions range over the positive rationals or the positive reals.

Alternatively, start by defining metrics and limits even more abstractly by saying that the metric should take values in some ordered field with the Archimedean property. Since $\mathbb Q$ is such a field, you can get started, and eventually it will turn out that $\mathbb R$ is also such a field, so you can reuse what you did from the beginning.

The "aesthetic" downside of the latter route is that it will turn out that all ordered fields with the Archimedean property are actually (isomorphic to) subfields of $\mathbb R$, so all that generality will not have bought you anything in the end. Thus people don't usually care to present the theory of metric spaces in that way.

Answer 2

One way of defining the real numbers is to say that they form an ordered field in which every non-empty subset which is bounded above has a least upper bound.

That then begs the question of whether such a field actually exists. The construction using Dedekind cuts, or alternatively using Cauchy sequences of the rationals, is sufficient to prove existence, and in fact then the least upper bound property does what you need.

There is also the question of whether such a field is unique, and indeed it is.

It is a common feature of mathematical definitions that a property is defined, and then is shown to be equivalent to some other useful properties and also to imply others. There is work to be done whichever equivalent definition is chosen, and sometimes we choose the definition which is easiest to work with in the context we have in mind.

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So it depends a bit on whether you are using limits to define the real numbers, or to construct them, and then show that the numbers you construct have the properties you need.

If you define the real numbers via Cauchy Sequences of the Rationals, then there is a logical step to show that Cauchy Sequences of the Reals behave as expected (as Henning Makholm has indicated). Whichever way you do it some care has to be taken - it is not as easy as it might at first appear to show that real numbers constructed as Dedekind Sections obey the usual rules of arithmetic. However, the details have been worked through, and efficient proofs are known. But usually much of this detail is taken for granted.

Answer 3

Here's, in modern language, how Cantor defined the real numbers.

A sequence of rational numbers $(q_n)$ is said to be Cauchy if, for every positive rational $\varepsilon$, there exists $k$ such that, for all $m,n>k$, $|q_m-q_n|<\varepsilon$.

A sequence $(q_n)$ is a zero-sequence if, for every positive rational $\varepsilon$, there exists $k$ such that, for all $n>k$, $|q_n|<\varepsilon$. Every zero-sequence is Cauchy.

Fact 1. The set $\mathscr{C}$ of Cauchy rational sequences forms a commutative ring under the operations $(q_n)+(r_n)=(q_n+r_n)$ and $(q_n)(r_n)=(q_nr_n)$.

The proof of this consists in showing that $\mathscr{C}$ is a subring of the ring of all rational sequences with componentwise addition and multiplication.

Fact 2. The set $\mathscr{Z}$ of zero-sequences is a maximal ideal of $\mathscr{C}$.

The proof is direct verification.

Fact 3. The field $\mathbb{Q}$ of rational numbers can be embedded in the quotient ring $\mathbb{R}=\mathscr{C}/\mathscr{Z}$, which is a field as well.

The embedding consists in associating to a rational number the corresponding constant (Cauchy) sequence and verifying that in this way we get a ring homomorphism.

If $q=(q_n)$ is a Cauchy rational sequence, I'll denote with $\hat{q}$ its image in the quotient ring $\mathbb{R}$.

Fact 4. The field $\mathbb{R}$ can be ordered.

Define $\hat{q}<\hat{r}$ if (and only if) there exists $k$ such that, for all $n>k$, $q_n<r_n$. It is readily seen that this is not dependent on the particular representative chosen for $\hat{q}$ and $\hat{r}$. Also the properties of ordered field are easy to verify.

Fact 5. (Cantor-Dedekind property) Every nonempty upper bounded subset of $\mathbb{R}$ has a supremum.

This is the trickiest part. A closed interval is a subset of $\mathbb{R}$ of the form $[\hat{q},\hat{r}]=\{\hat{p}\in\mathbb{R}:\hat{q}\le\hat{p}\le\hat{r}\}$ (where $\hat{p}<\hat{r}$). A nested chain of closed intervals is given by two sequences $(\hat{p}_n)$ and $(\hat{r}_n)$ such that

1. for all $n$, $\hat{p}_n<\hat{r}_n$;
2. for all $m,n$, if $m\le n$ then $\hat{p}_m\le\hat{p}_n$ and $\hat{r}_m\ge\hat{r}_n$;
3. for every positive rational $\varepsilon$, there exists $k$ for which $\hat{r}_k-\hat{p}_k<\varepsilon$.

It is not difficult (albeit tedious) to prove that, denoting by $\pi_n$ the $n$-th term of a rational sequence representing $\hat{p}_n$, and similarly defining $\rho_n$ with the upper bounds, then

1. $\pi=(\pi_n)$ and $\rho=(\rho_n)$ are Cauchy rational sequences;
2. $\hat{\pi}=\hat{\rho}$;
3. $\hat{\pi}=\hat{\rho}\in[\hat{p}_n,\hat{r}_n]$, for every $n$.

In simpler words, given a nested chain of closed intervals, there exists a (necessarily unique) element of $\mathbb{R}$ that belongs to every interval in the chain.

Deriving from this the Cantor-Dedekind property is standard.

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As you see, the above construction of $\mathbb{R}$ never uses the concept of limit, but it is of course modeled on it. Now it is essentially trivial to show that a Cauchy sequence in $\mathbb{R}$ is convergent (with the standard definitions).

The construction of $\mathbb{R}$ with Dedekind cuts allows for an easier derivation of the Cantor-Dedekind property. On the other hand, defining the field operations and verifying that we indeed get a field is much more boring. With the Cantor construction the field operations come almost for free from basic ring theory, but we pay a price when showing the Cantor-Dedekind property.

Both constructions give “the same” object, because it can be shown that two ordered fields having the Cantor-Dedekind property are isomorphic. The order relation on such a field is unique (essentially because every positive element has a square root). Such a field also contains a copy of the rational field $\mathbb{Q}$ and is Archimedean.

Thus it is not relevant how we build the real numbers. Once we show that there exists a construction of an ordered field satisfying the Cantor-Dedekind property, we can free ourselves from the details of the construction and only use the properties of this field.

Now defining metric spaces with a metric taking its values in $\mathbb{R}$ is not a problem any longer.