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November 19

Indescribable real numbers?

Consider the Real Number system and some of its natural subsets such as integers and rational numbers. We know that the set of integers is infinite, but countable. Also, rational numbers have been demonstrated to be countably infinite. Now consider the subset of real numbers that can be exactly described using some symbolic representation, such as the root of a polynomial, the sum of an infinite series, the limit to some equation, or some yet-to-be-invented notation (think integrals before Newton and Leibniz). Is there a name for such numbers?

In any case, I would expect that such numbers could also be countably infinite. The notational system would be countable, although not bounded by current state-of-the-art, and the combinations of these symbols of arbitrary length would also be countable infinite. I assume that symbolic representations can be linearized in a spirit similar to how mathematical formulas are converted Wikipedia <math> expressions.

I'm no mathematician, but can envision counting the possible symbolic representations as follows:

Count all combinations of maximum length 1 using only the first symbol in your vocabulary (total sequences = 1)
Continue counting with all combinations of maximum length 2 using the first two symbols in your vocabulary, excluding those already counted (total sequences now up to 6)
Continue counting with all combinations of maximum length 3 using the first three symbols in your vocabulary, excluding those already counted (total sequences now up to 39)
And so on (for all combinations of maximum length N using the first N symbols, count = $\sum _{i=1}^{N}i^{N}$ , I think)

Of course most such sequences would be gibberish, and some would represent duplicate values, but the point is that every valid symbolic representation that describes a number could be assigned an integer once we defined and sequenced our vocabulary.

Now, what's left is the concept of real numbers that can never be described. Is there a term for these? If we limit ourselves to pure mathematics (and exclude real-world measurements such as the angular momentum of the earth), do such numbers really exist? Can you prove that they exist? Can you name one? Do you just take it on faith that they exist?

(I would wager to guess that someone has thought this before, but just in case this is a major philosophical breakthrough, you heard it first from me here on Wikipedia!) -- Tcncv (talk) 02:25, 19 November 2008 (UTC)[reply]

So first of all, we have an article, definable real number. I have to warn you that the article is quite bad. And I say that as one of its main authors. This is a very difficult topic to write about while remaining faithful to WP standards (particularly WP:NOR).

The problem is that, yes, informally, it's clear that there are such numbers. But it's very difficult to say anything mathematically precise about them, including even giving a precise definition of what you're talking about. What happens in practice is that there are more- and more-powerful techniques for making definitions of reals, and if you specify one of them, then you can definably give an example of a real that is not definable according by that technique. However, to do so, you'll have to find a more powerful technique. The problem comes when you want to talk about real numbers that are not definable in any way at all, whatever that means exactly.

Related philosophical paradoxes are Berry's paradox and the paradox of the first undefinable ordinal number (can't remember what that one's called, if it has a name). --Trovatore (talk) 02:54, 19 November 2008 (UTC)[reply]

Wow, ain't Wikipedia great? Not only do I get an answer after only a few minutes to a really of-the-wall question, but it comes with a link to an existing article! Unfortunately, I need to do some studying of the article and brush up on my set theory before I can even attempt to convince myself that undefinable real numbers do in fact exist. Thank you Trovatore. -- Tcncv (talk) 03:22, 19 November 2008 (UTC)[reply]

Hi, Trovatore, IMVHO the problem of the first undefinable ordinal — at the first glance — must have something to do with the interesting number paradox. --CiaPan (talk) 06:42, 19 November 2008 (UTC)[reply]

There are some commonalities, yes. --Trovatore (talk) 08:30, 19 November 2008 (UTC)[reply]

Actually, there are some subtleties here that need to be mentioned. Note first that the set of all formulas in the variable x (i.e., set-theoretical "statements" about x) can be numbered in an effective way. As Trovatore points out, it is not possible to write down a formula F(n,x) meaning "object x satisfies the nth formula." So the only way that the idea of being definable by the nth formula can make sense, is if we consider that we have before us a model M of set theory which is itself a set, and n refers to an intuitive integer. By this I mean what we call integers in our own set-theoretical universe, not the objects in M that M considers to be integers, from its point of view. With the problem being seen this way, there is absolutely no argument given above that proves that there exist undefinable real numbers. This is because what M considers to be the set of real numbers could well be countable itself, from our point of view. See Skolem's paradox. In this case, the injection f (that we see) from M's set of real numbers to our set of natural numbers (which we identify with a subset - from our point of view - of M 's set of natural numbers), simply isn't a function that exists in M. In fact, although I don't have a proof of this, I think it's plausible that there exists a model of set theory in which every element is definable. Perhaps somebody better than I am at set theory can offer sufficient conditions on a model M of set theory for every one of its elements to be definable. It may be that it's simply plausible that such a model exists, but that this is not susceptible of proof because of Gödel's incompleteness theorem. 67.150.252.248 (talk) 09:08, 22 November 2008 (UTC)[reply]

Again thank you all for your input. Unfortunately, I don't think have a deep enough background to comprehend all of the set theory discussions and related articles. However, I have been looking at the Cantor's first uncountability proof article and in particular the theorem section. What I don't understand is how condition 4 is true for reals not for rationals. Maybe my thought processes are not abstract enough. Would anyone care to try to teach an old dog a new trick? -- Tcncv (talk) 00:30, 25 November 2008 (UTC)[reply]

Responding here to 67.150.252.248: I think it's a bit of a leap to say that the only way to make sense of the idea of definability is to relativize to a model. There is no first-order formula that expresses the idea "x satisfies the nth first-order formula"—that's true. But that just means that first-order logic is too weak to express the idea, not that the idea itself is in any way ill-specified. What is ill-specified, I think, is the notion of "well-specified definition". We can give examples of well-specified definitions, but we can never demarcate the borderline. However wherever the borderline is, it's certainly beyond first-order logic.

As to the specific technical question of whether it's possible for a model of ZFC to have every element definable within the model, the answer is yes. For example, take M to be the (Mostowski collapse of the) Skolem hull of the empty set inside L (Gödel's constructible universe). But that's just a model of ZFC, and a countable one at that. It doesn't really have all the real numbers; it just thinks it does. There are certainly real numbers that are not definable in the language of set theory by a first-order formula. It so happens that none of them are in M. --Trovatore (talk) 08:28, 25 November 2008 (UTC)[reply]

What is the generalized term for exponentiation?

By convention, if an associative binary operation is called "multiplication", the repeated application of the operation on the same operand is called "exponentiation". What is the term that corresponds to exponentiation if the operation in question is just some abstract $\diamond$ ? Alternatively, what is the term for the functional f, where

f:(\diamond :A\times A\rightarrow A)\mapsto (g:A\times \mathbb {N} \rightarrow A)

and g is the "exponentiation" for $\diamond$ ? —Preceding unsigned comment added by 98.114.146.90 (talk) 13:15, 19 November 2008 (UTC)[reply]

I assume you already read Exponentiation#Generalizations of exponentiation? For other extensions of the exponentiation concept, see tetration (which, admittedly, repeats a non-associative operator) and related articles. I don't really know of a name for that functional, though. -- Jao (talk) 15:00, 19 November 2008 (UTC)[reply]

Statistics - mtcars dataset

Hi, sorry if you don't think this question belongs here but i felt this is the nearest i could get.

I've just started playing around with R, the statistical program, and one of my first assignments involves a dataset called mtcars, which after looking on the internet seems like a commonly used dataset. One of the questions is to suggest a response variable and decide which are possible explanatory variables. Its just that i don't even know what some variables are. The ones listed are:

names - Name of the car
mpg - Miles per gallon
cyl - Number of cylinders
disp - No sure about, my guess is that its size of the fuel tank. Its a continuous measurement and the help page says the units are cu.in, which i assume is cubic inches.
hp - Horse Power
drat - The help says rear axle ratio, but i still don't understand what it is.
wt - Weight
qsec - Time to travel 1/4 of a mile
vs - It takes values of 0 or 1, and help says its V/S which is no help!
am - Automatic/Manuel
gear - Number of gears
carb - Number of carburetors
(Bold ones are ones i'm unsure of)

The question ask which is the obvious response variable, i'd have to say MPG as this is easiest to model, is this right? Then asks which are possible explanatory variables, my guess would be:
Carb - because it determines how much fuel enters the engine
Gears - Higher gears use less fuel, so cars with higher gears will have a better mpg
AM - Automatics cars may be more efficient changing gears than manual drivers
Wt - Weight of the car will alter the mpg
Disp - If this is the size of teh fuel tank, then wont it add more weight to the car which will make it require more fuel to move?

Thanks for any help
Pete
82.32.212.250 (talk) 19:15, 19 November 2008 (UTC)[reply]

"Disp" is presumably "displacement", the total volume swept out by the pistons. Other things being equal, more displacement means more power but worse fuel economy. --Trovatore (talk) 20:38, 19 November 2008 (UTC)[reply]

This link defines drat as "Rear axle ratio" which means drive shaft to rear wheel ratio. Not much help with vs however. -hydnjo talk 02:51, 20 November 2008 (UTC)[reply]

Just a guess but V/S could be V engine or Straight engine i.e. V/S = 0/1. -hydnjo talk 04:40, 20 November 2008 (UTC)[reply]

I found this, which didn't help with vs, though. In cars, acceleration enrichment (which we don't have an article on!) can be measured in volts per second (V/s). As your variable, "vs", can only be 0 or 1, though, I think hydnjo may well have it right. fish&karate 11:43, 21 November 2008 (UTC)[reply]