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When the irrational numbers were invented/discovered, it was said the number line was "complete." Before that, I'm sure everyone thought all the numbers in the world were rational and their number line was "complete" as well. (And they knew it was dense as well, meaning between every rational $p$ and $q$ there exists another rational $r$.)

Well, What makes us so sure $\mathbb{R}$ is complete? The fact that all the "holes" between the rational numbers were filled in? Why can't there be more holes in the Real line, holes that we haven't discovered yet, that other types of numbers can fill in? How can we be so sure there aren't more numbers in-between the Real numbers?

Perhaps solutions to hyper-exponential (tetration) equations like $x^{x^x}=2$? (Alas, $x$ is probably real and likely irrational.)

Of course, the Complex numbers extend the Real numbers, but not in the same way that the Reals extended the Rationals, which extended the Integers, which extended the Naturals. In the first three extensions, everything stayed in one dimension. In order to introduce complex numbers, we had to add another dimension, and of course with quaternions, etc, we have to add more dimensions.

But when it comes to one dimension, are we done? Are we positively sure we've accounted for everything once the Real line was constructed? Didn't we miss anything?

Edits, for those who ask:

The construction of $\mathbb{R}$ shouldn't matter, but the one I'm familiar with is the one that uses equivalence classes of Cauchy sequences of rational numbers, so let's use that one.

The definition of "complete" that might be easier to go off of is the least upper bound property: "every nonempty subset of $\mathbb{R}$ that is bounded above has a least upper bound." Equivalently, and my more favorite definition, is the nested interval property: "every nested sequence of closed intervals has a non-empty intersection," but this one might be harder to use.

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closed as not a real question by Andres Caicedo, Hagen von Eitzen, Davide Giraudo, Thomas, Cameron Buie Jan 6 '13 at 15:47

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Words in mathematics have precise meanings, and their validity is not just assumed. Under various precise definitions of "complete", the set of real numbers is complete. I recommend looking up some of these definitions. –  Jonas Meyer Jan 6 '13 at 7:49
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@TestSubject528491: So you are asking, if we construct the real numbers via equivalence classes of Cauchy sequences of rational numbers (including its order structure), then are we "sure" that the set of real numbers with this order structure satisfies the least upper bound property? Yes, and you can look up proofs in many books. –  Jonas Meyer Jan 6 '13 at 8:03
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@TestSubject528491: A definition means whatever we say it means. It will be useful for some purposes and not others. What do you want in a "legitimate" or "correct" definition? Without knowing what your criteria are, this seems subjective/unclear. (I do not think that speculating on how Archimedes would have used a modern mathematical word will be productive in this context, in part because we use definitions (in analysis at least) in a way foreign to pre-19th century mathematicians.) –  Jonas Meyer Jan 6 '13 at 8:09
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@TestSubject528491: What is the basis of saying that this property was considered by Archimedes to make the set of rational numbers complete? I find this dubious, but also don't see the relevance. Regarding the end of your last comment, what does "contains everything" mean? –  Jonas Meyer Jan 6 '13 at 8:14
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@TestSubject528491: You already know that the reals don't contain "everything". What is your definition of "everything"? –  wj32 Jan 6 '13 at 8:14
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5 Answers 5

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We know that the real numbers really are complete because we have a definition of what precisely does complete mean and moreover we have a proof that the real numbers satisfy that property. There are several equivalent ways to formulate what completeness means. One way to define completeness is by saying that every Cauchy sequences converges. Another is by specifying the least upper bound principle, and yet another is the greatest lower bound principle.

The real numbers can be constructed in various ways and for whatever way you choose there is a rigorous proof that the reals thus constructed are complete.

An intuitive way of thinking about completions (especially the construction of the reals by Dedekind cuts) is as a process of fillin in the holes. This is only intuition but it is a strong and useful one. It should be noted that the reals can, in a sense, be "completed" again by filling in holes. This time the holes are really "small" and the filling is done by infinitesimals. One constructions of such a "completion" is by using ultraproducts, yields the hyperreals ( http://en.wikipedia.org/wiki/Hyperreal_number). Since this is not a completion in the existing sense of the word this process is called 'enlargement'. It is interesting to note that one can start with the rationals, enlarge these via an ultraproduct construction, and then obtain the reals as a quotient of a certain subset. Thus, the completion of the rationals, yielding the reals, can be seen as a by-product of the process of enlargement.

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A metric space is called complete if every Cauchy sequence converges. This is a formal definition and roughly means that if we get "close" to a point on the real line, there is a "number" there that we are getting "close" to.

However, this does not mean we cannot come up with equations that have no solutions. For example $x^2 + 1 = 0$ has the solutions $\pm i$ which is an imaginary number. And in fact we can extend the real numbers to the complex numbers which has such a solution. However, in terms of completeness it doesn't add anything because there is no sequence of real numbers which get "close" to $i$.

Here are some number systems that might be of interest to you, but I do not know much about them
http://en.wikipedia.org/wiki/Hyperreal_number
http://en.wikipedia.org/wiki/Surreal_number
They add in infinitesimal numbers between real numbers. This does not contradict completeness though because we just need a Cauchy sequence to converge to something, and the real numbers provided that something. These other number systems may just provide "something else". Both articles would be an interesting read for you I'd imagine.

Edit: Be careful when mixing formal definitions with intuitive definitions. Completeness does not mean there is nothing to add, it is just a statement about convergence of Cauchy sequences.

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It seems to me that your last edit is the most relevant to the OP's question. (+1) –  robjohn Jan 6 '13 at 19:59
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The surreal numbers are in a sense a generalization of Cauchy sequences that "complete" any ordered field. For example, we have the surreal number $\{0.9, 0.99, 0.999, ... | 1\} = 1-\frac{1}{\omega}$ which is strictly between $1$ and every real number less than $1$.

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What's the difference between Surreal numbers and Hyperreal numbers? –  chharvey Jan 15 '13 at 2:47
    
@TestSubject528491 math.stackexchange.com/questions/221334/… –  Mark S. Feb 7 '13 at 23:32
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@Dan Brumleve, on the other hand, they fail the completeness property of the reals in the most extreme way: every nonempty set of Surreals has an upper bound and no least upper bound. But they're more "complete" in the sense that for any two sets A and B of Surreals where A<B, there is a surreal strictly between A and B. –  Mark S. Feb 7 '13 at 23:35
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The real numbers were used quite a long time before they were fully understood.

It wasn't until the 19th Century that the real numbers were properly defined and constructed as the "completion" of the rational numbers.

However this word "completion" has a precise meaning, only intuitively it means "filling in holes".

The true definition is that all Cauchy sequences converge in $\mathbb{R}$, whereas in $\mathbb{Q}$ this doesn't always happen (there are rational Cauchy sequences that converge to irrational numbers). In inventing irrational numbers we make every Cauchy sequence converge to a limit.

The way we invent these numbers from the rationals can really be viewed as an algebraic construction.

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It sounds like you should look up Dedekind cuts. This addresses on of the several notions of "Completeness" on the real numbers. Basically, Dedekind cuts define real numbers as the gaps between rational numbers. Any gap between two other gaps would be associated with another real number, and therefore there are none of these "gaps" between real numbers. This is the first notion of completeness, called Dedekind Completeness. Then, there is Cauchy Completeness. A sequence $\{a_n\}_n=0^\infty$ is Cauchy if:

For each $\epsilon >0$ there exists an $N$ such that $m,n>N$ implies $|a_m-a_n|<\epsilon$

A metric space is Cauchy Complete if all Cauchy sequences are convergent. This turns out to be true for the Real numbers. There is also a third kind of Completeness for the real numbers. This is often called the completeness axiom. It states that if a set $X\subset\mathbb{R}$ is bounded and nonempty, then it has a Least Upper Bound (note this is also called the Least Upper Bound property). Although this is an axiom, the real numbers woulddn't really be anything like what you saw in grade school without it. Right now, you may be wondering why the last two examples of Completeness are called Completeness. That is because it turns out that these three properties are all equivalent; each implies the other two. This is a very strong property, and it is hard to see Real Analysis, or even math in general, being the same without this Completeness of the reals.

On another note, you seem to have hinted at a notion of algebraic completeness of a field, which is generally called solvability. This has nothing to do with tetration in general, but rather the solutions of any number of equations existing. However, the real numbers aren;t "complete" in this aspect, even with polynomials. It is well known all polynomials are solvable in $\mathbb{C}$. However, this property, rather than being Real Analysis, is part of Abstract Algebra, more specifically, Field Theory. For example, the equation $x^2+1=0$ isn't solvable in $\mathbb{R}$. Also, this "tetration" example isn't generally solvable either; there are no solutions to $x^x=0$. Furthermore, not even all equations are solvable in $\mathbb{C}$. For example, there is obviously no solution to $\Re (z)=i$.

These are some interesting questions, and I'm glad to help. I hope the length of this answer isn't overwhelming. Best of luck in your mathematical pursuits.

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the lub (glb) property for the reals requires the bounded sub-set to be non-empty. –  Ittay Weiss Jan 6 '13 at 8:31
    
yeah, sorry. i'll fix that. –  dirty derwin Jan 6 '13 at 8:32
    
Thank you for your insight on solvability, it helped a lot. –  chharvey Jan 6 '13 at 23:12
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