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As it is true that we can construct all rational numbers $\Bbb Q$ from the set of integers $\Bbb Z$, is it possible to construct the set of real numbers $\Bbb R$ from $\Bbb Q$? If yes, how? Is there any procedure? And if no, is there any proof that we can't ? Thanks!

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In this answer you can find very detailed account of constructing real numbers using Cauchy sequences (i.e. as the Completion of the rationals with the usual metric): Completion of rational numbers via Cauchy sequences. –  Martin Sleziak Jul 6 '12 at 10:59
In this answer I've mentioned Artmann's book The concept of number, where several constructions of real numbers are compared. –  Martin Sleziak Jul 6 '12 at 11:11
There are two main methods: Dedekind cuts, and equivalence classes of Cauchy sequeences. You can easily look them up now that you have the magic words. –  Arturo Magidin Jul 6 '12 at 18:29
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4 Answers

up vote 6 down vote accepted

Consider the following set of rational numbers:

$S =\{x \in \Bbb{Q}:x < 0, \text{ or } x^2 < 2\}$

This set is clearly bounded above, for example, $2$ is an upper bound. The thing is, the set has no rational least upper bound. In fact, it seems like we can get quite a good idea of "how big" such a least upper bound would have to be: if we call it "$x$", you can see that: $1.4 < x < 1.5$, for example.

So, the general idea is to capture all the "magnitudes" that rational numbers can approximate, but never hit "exactly". There are a few different ways to do this: Dedekind cuts, Cauchy sequences and "infinite decimals" being the most popular. All of these are designed to do one thing: ensure that the least upper bounds of sets like $S$ always exist (equivalently: ensure that (rational) Cauchy sequences converge "to something").

This process is called "completion", and can be thought of as "filling the gaps" in the rationals (so we get a continuum). It is, at the heart of it, a topological construction, and the need for it doesn't become clear until one starts to investigate continuity. For algebra (that is, solving polynomial equations with rational coefficients), algebraic numbers would suffice (things involving square, cube and n-th roots, and the like), but analysis of functions often requires we get an estimate of "the size of something". Intuitively, we want such "sizes" to be bona-fide numbers, that we can manipulate according to the familiar rules we learn early in life (that is, we want an ordered field, at the very least).

So, yes, there are several such constructions...what is fascinating is that each construction gives us "the same object" (only the names are changed, to protect the innocent). This justifies the real numbers being called "THE" complete ordered field (although to be 100% correct, one should say "complete archimedean ordered field", because there are complete ordered fields which are not (isomorphic to) the real numbers, such as the hyperreals).

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They are not the same object under some commonly used logical systems, such as constructive systems. There is a rich theory of the relationship between the different constructions, and under which axioms they become identical. –  ex0du5 Jul 16 '12 at 19:33
@ex0du5 Are you sure that Cauchy sequences and "infinite decimals" come out different in "some commonly used logical systems"? I cannot imagine how this might be. –  MJD Jul 16 '12 at 19:41
@MarkDominus: I may have parsed that sentence differently than you, with the "Dedekind cuts" as one construction and "Cauchy sequences and "infinite decimals"" as another. Those are definitely different in a number of systems (those that do not have weak countable choice and related connections). I did not expect that David would be distinguishing Cauchy sequences from a particular form of cauchy sequences using rationals with denominators that are powers of 10. That seems a weird distinction to me. –  ex0du5 Jul 16 '12 at 19:49
@ex0du5 your comments led me to investigate "reverse analysis" of complete ordered fields, for which I am most grateful. –  David Wheeler Jul 19 '12 at 7:11
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The real numbers can also be constructed from the rationals via Cauchy Sequences.

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To complete the list from Spivak, we can also define the reals 'naively' as infinite decimal sequences, but this is essentially picking a representative element from an equivalence class of Cauchy sequences.

(eg $\pi$ should be the limit of the sequence 3, 3.1, 3.14, 3.141, ...)

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The set of real numbers is the complete ordered field containing the rationals which is unique upto isomorphism and in fact each real number is defined as set of rationals satisfying certain properties called Dedekind Cuts. See also the link @anon posted.

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"in fact each real number is defined as sequence of rationals" is very misleading. –  Michael Greinecker Jul 6 '12 at 10:25
I just tried to say that real numbers are defined using rationals in layman's language –  pritam Jul 6 '12 at 10:30
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