# Are there more real numbers than we can actually imagine?

I mean, if we could imagine all the real numbers then we could assign each number a finite sentence (or a finite book).

Since the set of the finite books is countable then the set of real numbers would be countable too.

Is this correct? Is $\mathbb{R}$ (made by our thinking) larger than our own thinking?

By imagine I mean to describe.

• There is a good point made in the post. We have a rather incomplete grasp of $\mathbb{R}$, and indeed of the power set of $\mathbb{N}$. – André Nicolas Nov 5 '13 at 5:41
• Can you describe everything you can imagine? Have you never thought you'd something which was ineffable, beyond any words? – Asaf Karagila Nov 5 '13 at 5:42
• As Han Solo once said, "I dunno. I can imagine quite a bit." – David H Nov 5 '13 at 5:44
• My imagination is pretty complex. Even if it turns out that can could describe each individual thing I can imagine, I don't think I can imagine a whole function that maps descriptions to my imaginings. – user14972 Nov 5 '13 at 6:18
• Some would find it contentious that $\mathbb{R}$ is "made by our thinking". Some would also find it contentious that "imaginings" or "thoughts" are individuated enough to sensibly number. People like me do both :p – Malice Vidrine Nov 5 '13 at 6:37

Not only there are more real numbers than we can describe, there are more natural numbers than we can describe.

Our abilities to describe somethings, while virtually unlimited, are quite limited. We are bounded by time, by space, by the lexicon of our language. We are bound by mortality. It follows, if so, that the largest number that you can even begin to describe, is quite small (that is, almost all the other numbers are larger).

Taking this to a somewhat more formal level, if by "describe" you mean define somehow using a first-order formula in a language including addition, multiplication, elementary functions, integration, predicates for "known sets" (e.g. the integers, and so on), the familiar constants (e.g. $\pi$, $e$), and all those things that you know... you could still only describe only a very few numbers, i.e. a countable number of them. Since there are uncountably many real numbers, we have shown that almost all the real numbers cannot be described.

But even then there are caveats about what do we mean by the word "define". There are models of set theory in which every element is definable without parameters. Including every real number. There are subtle things to mind here, and if you are interested, then I suggest you to read (at least the well written introduction) the following paper:

J. D. Hamkins, D. Linetsky, and J. Reitz, “Pointwise definable models of set theory,” Journal of Symbolic Logic, vol. 78, iss. 1, pp. 139-156, 2013.

And the arXiv version can be found here as well.

Keep in mind, though, that some understanding in logic is required to even approach this question seriously. You should have a good grasp on basic things like the difference between theory and meta-theory.

I would say you need to be careful what you mean by imagine.

Consider a rainbow in which we see all of the visible spectrum. Within this spectrum we can see infinitely many slightly different shades of colour. You can see all of them, they're right there as a physical manifestation, you're not "imagining" anything so to speak. Now could I sit down and colour in each page of a book, one of every single colour in that spectrum? No, I couldn't. I "know" what everyone of them looks like. If you give me a colour, I can say "well of course that's one of them, in fact you can see it around this region of the spectrum". However, there are simply too many, infinitely many, colours for me to actually draw out myself.

The real numbers are essentially the same. We know them, we can describe them and we have an intuitive and even physical feel for how they look, but alas, we can not write them all out because there's just too many. To me though, that's different from saying that I can't imagine them.

The issue here is with your idea of imagining: yes, we can imagine any particular real number. We can imagine any $x \in \mathbb{R}$. However, we cannot imagine any closed interval that is a subset of the reals: we can imagine the abstraction, but we certainly cannot imagine every individual number in that interval, because the number of reals in that interval is uncountably infinite.

Similarly, the idea of assigning a finite 'book' to every sequence of reals, or to a subset of $\mathbb{R}$ is inherently flawed, precisely because there are so many reals: it is one of the properties of the real numbers that between every two reals, there is yet another real. We could fill the entire universe with books listing the real numbers between $0$ and $0.1$, and we still wouldn't be done.

• Any given $x\in\mathbb{R}$. Not just any $x\in\mathbb{R}$, there's my point, there are numbers which can't be described in a finite formula or text, so can't be imagined. – Carlos Pinzón Nov 5 '13 at 6:18
• @CarlosPinzón such as? Name such a number. – Newb Nov 5 '13 at 6:27

Since you mention that "imagining" a number would let us assign to each number some object from a countable set, it seems to me that your question is just a veiled way of asking if we can list all of the reals (i.e., construct a bijection from the reals to a countable set). Which, of course, we can't do.

And if by "imagining" a number you mean "having a concept of/thinking about that particular number," then we certainly can't imagine all the reals. In this sense, we can't even "imagine" all the numbers of an infinite countable set, since we could never "think about" each particular number because there will always be a number in the set we haven't thought about yet. But this isn't really a rigorous mathematical concept like countability/uncountability.

To me the real numbers and the rationals are very much like analog and digital. You can try to turn the volume to $5.5$ out of $10$ by setting the maximum volume to $20$. But the fact of the matter is, as much as you try to refine the scale, you will never be able to get it to the point of the analog scale.

Another example would be like the uncertainty principle in physics. The moment you try to write $\sqrt{2}$ in decimal form, you lose its essence so to speak. (So, not very much like the uncertainty principle. :P)