Hello everybody, Yesterday I've read that there exist a real number r which cannot be defined by a finite number of words. This result, although quite awesome, is so strange that it lead Poincaré to doubt Cantor's work and state "never consider objects that can't be defined in finite number of words". In our course on analysis, our professor postulated the existence of a supremum for any bounded sequence of rational numbers and I think the existence of the set of real numbers followed. I am not entirely satisfied by this approach though, because it postulates supremum, which is not particularly simple notion. Do you think the set of real numbers can be defined in finite number of words, using only basic concepts? (whole numbers, simple logic, ...) Or do you think it is impossible and the set has to be postulated by force? What are your feelings about this? Which possibility do you prefer? Jano
There are many ways to show that the field of real numbers "exists" in the sense that a field with the desired properties can be defined from the field of rational numbers. The most popular ones seems to be Dedekind cuts (see e.g. "Set theory for guided independent study" by Derek Goldrei), and metric space completion (see e.g. "Foundations of modern analysis" by Avner Friedman, for a proof that metric spaces can be "completed"). I very much doubt that there could be a way that's much simpler than those. I like the metric space completion method, because it has other uses that are significant to physicists, so it's less likely to be a waste of time to learn it. These constructions clearly do use only a finite number of words. What your professor was talking about is how a field that's defined using one of those methods contains members that can't be specified in a finite number of words. The argument is simple: Every real number has an infinite decimal expansion (possibly ending with infinitely many zeroes). If we can specify that decimal expansion in a finite number of words, we can write a computer program that generates all those decimals, one at a time. The program may never finish its run, but the program itself is a representation of that number. The problem is that there's only a countable number of computer programs. This implies that a number that can be generated this way belongs to a countable subset of ℝ, but ℝ is uncountable, so that subset can't be equal to ℝ.
Thank you Fredrik. I think what puzzles me is this: The set of real numbers is defined with finite number of words (say, 500 words). But there is an element of this set which cannot be defined with finite number of words (say, 1,000,000,000 words is not enough). It seems that one element of the set is more complicated than the set itself. This is very strange. I would believe it if it was exactly the opposite. Jano
Yes, it's kind of strange. It got even stranger when Max Tegmark used it to argue that a multiverse with infinitely many universes is a simpler idea than a single universe. Link.
Yes, this is true, there are real numbers that are "more complicated" than the set of real numbers. But it's quite unfair to say that this is a problem with real numbers only. The situation truly shows up everywhere. For example, the set of prime numbers are very easy to describe, but the seperate prime numbers can be quite complicated. This is just one example, but the more you progress in mathematics, the more that the situation shows up. It's generally easier to describe the set of elements than the individual elements.
The number of finite sets of words is countable. So they can not specifically define each real number since there are uncountably many of them one can define the reals from the rationals using the idea of Dedekind cuts. This require no other ideas than the euclidean metric on the rationals. I am not sure how ones extends the arithmetic to these without notions of Cauchy sequence. Maybe it is not possible.
Goldrei does this. A real number is defined as a set ##\mathbf{r}\subset\mathbb Q## such that (a) ##\mathbf{r}\neq\emptyset,\ \mathbf{r}\neq\mathbb Q## (b) If ##p,q\in\mathbb Q## are such that ##q\in\mathbf{r}## and ##p<q##, then ##p\in\mathbf{r}##. (c) ##\mathbf{r}## doesn't have a maximum element. The definition of addition is simple: ##\mathbf{r}+\mathbf{s}=\{p+q|p\in\mathbf{r},\ q\in\mathbf{s}\}##. Multiplication is a bit trickier, but it doesn't look too hard. The hard (tedious) part is to verify that all the axioms of the real number field are satisfied.
The crux of the issue is the step where we define R to be the set of all dedekin cuts (or cauchy-sequences), without specifying each dedekin cut we are considering. Set theory allows us to do this through its axioms, some of which Poincare undoubtedly would have dismissed on the same token. To be fair, it's not a meaningless criticism, but almost all mathematicians today accept set theory, or specifically ZFC, in mathematics.
Thank you Fredrik! That is exactly the kind of definition I was thinking about. Clear and short. But still it is suspicious that you define real number and the set of real numbers at the same time. Is it possible to disentangle them? (First real number, then the set). Disregardthat, what would be the disputable step in this definition for Poincaré? Is it the fact that we are defining the whole set of real numbers without constructing its elements first?
Yes, the real numbers are the unique complete totally ordered field. http://en.wikipedia.org/wiki/Real_number Each of those technical terms (complete, totally ordered, field) can be defined in terms of logic and set theory. It's not mysterious that a set is simpler to describe than its elements. If you have a bag of groceries, then "bag of groceries" is its description. But a tomato has a complex molecular and physical structure that's not described by "bag of groceries." Sets are often simpler than their elements.
Steve, I agree there is nothing mysterious with the bag of groceries. You already know what grocery is, so you can imagine some of your favourite vegetables instead of "groceries". But for reals it is kind of peculiar, because we are defining a very special set of elements and we do not have the definition of these elements in hand. It is like "bag full of krandak", where you (and I) do not know what krandak is.
This definition already does it that way. The definition says that a subset of the rational numbers is said to be a real number if it's closed from the left (i.e. satisfies condition b), doesn't have a maximum element, and isn't equal to either ##\mathbb Q## or ##\emptyset##. Edit: The justification for this is that one of the ZFC axioms says that given a set S and a property P, there's a set whose members are precisely the elements of S that have property P.
Ah, of course you are right, you define what the real number is and then the set of them follows. My last post was silly. Thanks, I need to think about this more.
This looks like you could end up with one of those logical paradox puzzles. For example: "Smallest positive number than cannot be defined by a finite number of words" defines that number with a finite number of words!
That is an interesting example. First it seems like it refutes the existence of those weird numbers. But I were on the other side I could still say that it is only your definition of the number which is inconsistent - there is no smallest positive number, because there is an infinite number of them (arbitrarily close to 0). But I like the example. Maybe it can be improved somehow? Thinking about these weird numbers, everytime you would find such a number, you would contradict its property that it is undefinable in finite number of words. Seems to me much like claiming the existence of non-existing things.
I don't believe there is any positive integer that can't be described in a finite number of words. So the set you've defined is empty, and has no smallest member. Any finite integer has an English-language name -- "eleven billion, eight hundred seventy million, six hundred two thousand, and 47" is the English language name for a particular number. Every number has such a name, as long as we're allowed to keep making up names for the powers of 1000, like thousand, million, billion, trillion, humonga-gazillion, etc. Currently only finitely many of these are defined. But even this is not a problem. If you didn't have a name for a million, a thousand thousand would do. So in fact we can just say that "one thousand thousand thousand thousand" stands for 10^12, whether there's a name for it or not. With that convention, every positive integer can be named in a finite number of words. I believe the original paradox is "the smallest positive integer that can not be named is less than 1000 syllables." Now THAT defines a particular positive integer which we just named or characterized in less than 1000 syllables. So that's a valid paradox. To repeat: "the smallest positive number than cannot be defined by a finite number of words" does not exist; because every positive number (assuming you mean integer) can be described in a finite number of words.
although as others have pointed out, "finite number of words" is a poor criterion, similar (and more strongly paradoxical) formulations are possible. you might enjoy reading this: http://en.wikipedia.org/wiki/Berry_paradox under certain circumstances, this is equivalent to Godel's Incompleteness Statement, that there are (in certain formal systems) true, yet unprovable statements. the "root cause" of most of these thorny questions, is in "defining "definable"". it's hard to avoid self-reference in giving a definiton of "definition". one way this is avoided, is to create separate "layers" of meaning, and only talk about "higher layers" from "lower layers". so, for example, you can talk about the truth or falsity of a sentence, as long as that sentence itself isn't asserting it's own truth or falsity, which neatly side-steps the "this sentence is false" paradox as "ill-formed". perhaps you can see that doing this, is going to require a "lot" of layers, and keeping them all straight makes for some complicated bookkeeping.
If you take the set of positive reals that are not expressible in a finite number of words, what makes you think that set has a smallest element? The set is bounded below by zero, so at best we can say that the set has an inf. I suppose you could fix this up by saying that your paradoxical number is the inf of all the positive reals not expressible etc. Is that what you had in mind?
By the Well Ordering Principle, the set can be well ordered. Such a set has a first element under the well ordering.