[chowells]

What are "real numbers"? A horribly misnamed fiction. Nearly all of them cannot be represented with a finite amount of information. I strenuously object to naming an uncountable set "real" when only a countable subset (measure 0 of the full set) can be worked with in any way at all.

We need to stop venerating the "real" numbers and start focusing on sets that are actually usable.

[flebron]

This is a similar argument to sqrt(2) being "not a number", back in the BC's, because it was not rational. And yet, you can construct it in a straightforward manner by making a right angled triangle with catheti of length 1, giving a hypotenuse of length sqrt(2). I suppose this would have made you equally uncomfortable back then.

One can definitely "work with" numbers that aren't easy to write. a + (-a) = 0, and this is valid for every real number a, not just "the ones which I can describe with a finite amount of information", or the ones I've written down at some point in my life.

[gizmo686]

The 'problem' with the reals is that there are numbers that cannot be constructed.

Every number that we can construct can be constructed in a finite amount of symbols. For example sqrt(2) is an unambiguous description of itself. Without use of the sqrt function, we can also call it the number x such that x*x=2. However, every description is a finite string constructed from a finite alphabet. We can easily show that the set of all such descriptions is countably infinite. However, we can also show that the set of all real numbers is uncountably infinite. Therefore, there is an uncountable infinity of real numbers that cannot be constructed.

[ColinWright]

Indeed, and the constructable numbers are studied as a subset of the reals, as are the algebraics, and the computables. You can make a choice as to the domain of discourse. If you like, feel free to restrict it to the computables (or the constructables).

Then apply the diagonal argument. Take the computable numbers between 0 and 1, including 0, not including 1. These are countable, so we can write them in a list, taking a mapping k from the natural numbers: { 1, 2, 3, 4, ... } to the set of computable numbers in [0,1).

Now let's construct a new number. In the first decimal place we put 1 if the first decimal place of k(1) is 0, and 0 otherwise. In the second place we put 1 if the second decimal place of k(2) is 0, and 0 otherwise. And so on.

This results in a number that's not on the list, and is between 0 and 1. So it must, by our assumption, not be computable.

Things become tricky.

So there's a choice to be made, and most mainstream mathematicians have decided to talk about, use, study, and otherwise accept the existence of the real numbers because it's convenient.

Feel free to choose otherwise.

[gizmo686]

I can't find a flaw in your arguement, but it seems like it leads to a contradiction.

Let a constructable number be one which can be unambiguously described in a finite string. Because we are working from a finite alphabet, we can trivially see that their is a bijection between the constructables and the integers (if we have n symbols, then each string can be read as an integer in base n, so the amount of constructables is no larger than the integers. We can also show that all integers are constructable, so the amount of constructables is no smaller than the integers). Now, take the set of all constructables, and use the diagonal arguement to construct a new number. We can see that this number is not constructable, however, it would appear that I have just unambigously described it, meaning that it must be constructable.

The only potential hole I see is that the ordering of the constructables when I apply the diagonal arguement is ambigous, but we can unambiguously order them by the lexical ordering of their 'canocial' description, and we can unambiguous define the canonical description as the smallest one when translated into a base n integer.

I suspect that doing the above will run into problems with computable numbers (as it likely involves the halting problem), however it appears to be an unambiguous description of a real number that is not constructable. Obviously there is some flaw in this reasoning.

[jfarmer]

You're using too many imprecise terms. First, what does it mean for something to be able to be "unambiguously" described? As opposed to ambiguously described? You have to define it.

Second, what does it mean for something to be described (unambiguously or otherwise) with a "finite string?" What is a "string" here?

You're playing too loose with these ideas and it's biting you. You have to start by defining them precisely. For example, I don't see at all how the new number not on your list is "described unambiguously." It's presumably not enough to say "there is some number not on my list, we will call it x" since we know there is more than just one such number. How is that unambiguous?

In any case, that's why you have to define these things precisely.

[archgoon]

How do you know if a number is constructible? It is described by a computer program. Although the set of computer programs can be enumerated, determining if a program it will print out any digits is not something that cannot be determined. So yes, Halting Problem. :)

So you cannot list all constructables in a constructive manner because the list itself is not constructible.

A related concept

https://en.wikipedia.org/wiki/Chaitin%27s_constant

[arnodel]

I reckon the flaw is that your bijection between N and your set of "constructable" numbers is not itself "constructable". In fact your argument can probably turned into a proof that there is no such "constructable" bijection.

[lmkg]

The Reals are venerated because they're actually usable. Other numbers systems tend to be a gigantic pain in the ass to get any work done with.

The Reals are constructed specifically to be the smallest set that has some nice algebraic properties, like Least Upper Bounds. Sets that model the real world, like the constructables, countables, computables, etc. tend to be subsets of the Reals, and therefore don't have those properties. That absence makes life difficult.

The Real Number system, like almost everything in mathematics, is an approximation of reality that makes a trade-off between faithfulness and tractibility. As it turns out, gaining more of the former loses you quite a bit of the latter. It's generally not worth it.

[gizmo686]

Do you have any idea of what set we should use to replace them with? The rational numbers can do a lot, but we have discovered that there are numbers worth talking about (and which can be described) that are not rational. Whatever replacement you propose must be usable where ever we would use real numbers, and must be at least as simple to use.

[moyix]

One possible replacement is the computable numbers [1]; this includes the algebraic numbers and some common transcendentals (e, pi), and you can even build up something akin to standard analysis (computable analysis [2]).

[1] http://en.wikipedia.org/wiki/Computable_number

[2] http://en.wikipedia.org/wiki/Computable_analysis

[gizmo686]

Unfourtuantly, there exist numbers which are definable but not computable.

[moyix]

Sure. Chaitin's Omega is a good example. The question is whether such numbers occur in the real world.

[jfarmer]

The trick here is defining "occur" and "real world" precisely. Are you saying the act of me writing down those symbols and expressing the idea does not count as "occurring in the real world?" ;)

[moyix]

This reminds me of the self-defeating property of an "uninteresting" number -- a reasonable definition might be "any number that does not have any property of human interest", but then of course there is a smallest such number, and so that has the interesting property of being the first uninteresting number, a contradiction!

You're right though that a more precise definition of "real-world numbers" is needed, but I confess that my attempts to think of one in the past few minutes have been essentially circular (coming down to "the ones we know how to compute")!

[acjohnson55]

Unfortunately, the set of computable numbers, while countable, is more difficult to work with than the much more conceptually reasonable set of reals. Luckily, we have pretty great theoretical tools for dealing with uncountability, so I don't think the fact that the vast, vast majority of real numbers are unidentifiable is really that big of a problem. That, and the continuum is a really useful concept, even if it very well might not have any basis in physical reality.

[scotty79]

IMHO real numbers are anything but. I believe there isn't a single thing in the universe that is represented by real number. Any physical law that involves pi should be considered as statistical in nature. There are no perfect circles. Only the things that are really well approximated by them.

[twobits]

Amen to that. The real world is discrete. The real numbers in our equations are just approximations.

[dnautics]

imaginary numbers are also a horribly misnamed fiction. For decades, my dad was mesmerised by how if you plugged in a bigger than c value into the lorentz transformations (he had long since forgotten the form of the transformation) you would become "imaginary". To set him straight, I asked him, if instead we named them "Green" numbers, would you become "green" if you went faster than the speed of light?

[baddox]

What do you mean by "represented with a finite amount of information"? Are you referring to their representation in a positional notation like decimal or binary? Or are you referring to the much subtler and more advanced fact that almost all reals are uncomputable? The former isn't really true, and the latter, while true, is subtle enough that it doesn't matter for the vast majority of mathematics (and to replace the reals with the computable numbers would make most of mathematics messy).

[gizmo686]

I don't think "represented with a finite amount of information" means computable. For example, consider BusyBeaver(n). We have shown that their exists an n such that BusyBeaver(n) is uncomputable. However, "BusyBeaver(n)" still contains enough information to describe this number. However, because all descriptions are a finite string from a finite alphabet, we can show that only a countable infinity of descriptions exist. However there exist an uncountable infinity of real numbers. Therefore, most real numbers cannot be unambiguously described.

[baddox]

Again, it just comes down to what we mean by "information" and "description." We can certainly construct the real numbers using a finite amount of precise language, so it's reasonable to claim that we have described all real numbers. Heck, even the existence of the English phrase "all undescribable real numbers" evokes an interesting linguistic and philosophical debate, similar to http://en.wikipedia.org/wiki/Interesting_number_paradox.

[gizmo686]

We can construct the set of all real numbers with a finite amount of information. However, that set contains elements which we cannot precisely describe with a finite amount of information.

The phrase "all undescribable real numbers" does not introduce any problems, because we have still not described any specific undescribable number. We would run into a problem with a phrase such as "the smallest undescribable real number", as that would be a description of a specific undescribable real number. Fourtuantly, that particular phrase does not raise any problems because we can simply conclude that their is no smallest undescribable real number, in the same way that there is no smallest real number in general.

[GFK_of_xmaspast]

I don't see why we need to be shackled to the bounds of countability.