Constructing $\mathbb R$

Question

I am learning mathematical analysis. In one of the pages of a book on analysis I found a statement which I could not digest. The statement was "Cantor constructed $\mathbb R$ using nested intervals". Another such statement was "Dedekind constructed $\mathbb R$ out of $\mathbb Q$ using his ideas of Dedekind cuts".

Could you, my fellow users, kindly explain what is the meaning of this statement? Is "construction" having some indispensable mathematical definition? If the answer to the second question comes in positive then I will be grateful if somebody puts up the definition.

Answer 1

Construction does not have a formal mathematical definition. However, usually it means the proof of the existence of something by using permissible methods (which depend on the context). Sometimes people use the term construction in a sense more narrow than mere existence. For these people, construction means the proof of the existence of the object using some constructive methods that tells what the object is than merely verify that it has the desired property. (I don't know if this last sentence made any sense.)

As an example, you may have seen the rational numbers constructed as equivalence classes of integers. You can write down the axioms for the rational numbers, but how do you know that there is a model for that theory (i.e. is it consistent)? The familiar construction using equivalence classes produces a structure that satisfies all the axiom that you wrote down for the rational numbers. However, this may be considered a construction because you actually know more information. Because you know each rational number is actually a equivalence class of integers, you may actually be able to know more.

Actually whether you can prove more depends sometimes on what language and theory you are in. If you write the rational numbers or even real numbers merely as a dense linear ordering without endpoints in the language of linearly ordering (that is you ignore all its algebraic properties), you actually don't prove anything more. Everything you can prove in the model of the rational number, you constructed - even though you know everything is an equivalence class - you could have already proved from the DLO axioms. This is because DLO is complete.

However, something may be non-constructive to some may be the axiom of choice. Given a family $\mathscr{A}$ of nonempty sets. There exist a function $f$ from $\mathscr{A} \rightarrow \bigcup \mathscr{A}$ such that $f(X) \in X$ for all $X \in \mathscr{A}$. If you use the axiom of choice, some may consider this not constructive. Even though the axiom asserts the existence of the set you desired. You do not really know anything about $f$ other than $f(X) \in X$ for every $X \in \mathscr{A}$.

Answer 2

To say that we construct a mathematical object is to say that under some assumptions (axioms of some mathematical theory) we can define an object from another object whose existence was asserted earlier (either by assumption, or it may be provable from the theory we assumed before).

This is a somewhat an expansion of the idea of an algorithm. In the freshman year an algorithm is introduced as a "recipe" for cooking a certain result from starting parameters. A mathematical construction is similar, in a broader context. It is the method in which we define a new object from a parameter within a certain theory (which assert we may do some operations, e.g. take power sets, unions, etc.).

For example, we construct $\sqrt{2}$ from the rational numbers as the positive solution of $x^2-2=0$, of course one can argue that this is a number in $\mathbb R$ or in $\mathbb C$ and so on. However, without any knowledge of these objects I can prove that $\sqrt 2\notin\mathbb Q$ and I can also define $\sqrt 2$ as this number and show that we can have a new field, $\mathbb Q[\sqrt 2]$ which has $\mathbb Q$ and in which $x^2-2=0$ has a solution.

In the case of the real numbers, we can define them from $\mathbb Q$ (our parameter) in several ways. That is we define a mathematical object, using $\mathbb Q$ as a parameter, and this object will have the properties we want from $\mathbb R$.

The Dedekind cuts construction tells us that a real number $r$ can be defined as $\{q\in\mathbb Q\mid q<r\}$. Of course this is not a number per se, but rather a set, while rational numbers need not be sets for themselves (although they might be constructed as sets as well).

We can define addition and multiplication on these sets that work the way we hope them to work. Addition, for example, is defined as:

$$x+y=\{q+p\mid q\in x\land p\in y\}$$

Note that the left hand side is the addition over sets of rational numbers, while the set on the right hand side is a set defined using addition in the rationals which we already have.

The other properties of $\mathbb R$ soon follow as well. What sort of properties?

1. This is a field, with a total order which respects the operations of the field, i.e. it is an ordered field.
2. It has a copy of $\mathbb Q$ such that between two different real numbers there is a number which is from this copy. (Since we define the "numbers" as sets, we don't have the original field $\mathbb Q$ but rather an exact copy of it)
3. Every bounded subset has a least upper bound.

One can go through the list of axioms required in these properties (axioms of a field, etc.) and see that the object we define using Dedekind cuts indeed satisfies all of those properties.

Answer 3

"Construction" of the real numbers means showing there exists a set with all the properties of real numbers. We generally approach this in the setting of ZFC (axiomatic set theory).

One way to do it is to start with the empty set $\emptyset$ and call this set $0$. Then you define $n+1 = \{0, 1, ..., n\}$ so you now have the set $\mathbb{N}$ of natural numbers. The existence of this set follows from the axiom of infinity.

Now you construct $\mathbb{Z}$, the integers from $\mathbb{N}$. Basically represent each integer as the equivalence class of ordered pairs of natural numbers $(a,b)$ under the equivalence relation $\sim_{\mathbb{Z}}$ where $(a,b) \sim_{\mathbb{Z}} (c,d)$ if and only if $a+d=b+c$ (this property captures integer addition).

From $\mathbb{Z}$ you construct $\mathbb{Q}$ where each fraction $\frac{a}{b}$ ($b \neq 0$) is the equivalence class of fractions (aka ordered pairs of integers) equivalent to it under the relation $\sim_{\mathbb{Q}}$ where $\frac{a}{b} \sim_{\mathbb{Q}} \frac{c}{d}$ ($b \neq 0, d \neq 0$) if and only if $a \cdot d = b \cdot c$ (this property captures rational multiplication).

Now finally you construct $\mathbb{R}$ by considering Cauchy sequences of rationals. For example, $\sqrt{2}$ can be represented by the sequence $x_n$ of rational approximations to it (That is $x_1=1, x_2=1.4, x_3=1.41, x_4=1.414, ...$). You can then define $\mathbb{R}$ as the equivalence class of Cauchy sequences under the equivalence relation $\sim_{\mathbb{R}}$ where for Cauchy sequence sof rationals, $\{x_n \}, \{y_n\}$, that $\lim_{n \rightarrow \infty} x_n - y_n = 0$ (this property says if two Cauchy sequences converge, we consider them equivalent).