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What are the 'real numbers', really?

math.vanderbilt.edu

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Re: What are the 'real numbers', really?

#51

Post by gizmo686 »

ColinWright wrote: post #42
gizmo686 wrote: post #26

Earlier quoted context omitted.

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 t…

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, tak…

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.

Re: What are the 'real numbers', really?

#52

Post by gizmo686 »

baddox wrote: post #38
chowells wrote: post #6

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.

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…

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.

Re: What are the 'real numbers', really?

#53

Post by arnodel »

gizmo686 wrote: post #51
ColinWright wrote: post #42

Earlier quoted context omitted.

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, tak…

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 constructab…

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.

Re: What are the 'real numbers', really?

#54

Post by mcguire »

Someone wrote: post #37
igravious wrote: post #24

Earlier quoted context omitted.

Excuse me. Of course. I was using line and line segment interchangeably there. Which I should have not been doing if I am aiming for clarity but I think my point (ahem) applies to line segments and lines that extend indefinitely in one or two directions. Presumably people will contend that even a line segment "contains" an infinite number of points. But if points have zero extension then even an infinity of them cann…

"But if points have zero extension then even an infinity of them cannot sum to anything greater than zero." Infinities are weird; anybody who wants to learn math has to accept that. 0.99999… does equal 1, there are as many even numbers as integers, etc. these things are 'true' not because they make sense initially, but because they make the most sense of all the other things we have thought of so far. Similarly, a se…

"a set of Aleph-0 points can completely cover a line"

Aleph_0 is the cardinality of the integers. I don't think that'll cover a line. For that, you need the cardinality of the reals, C, which may or may not be Aleph_1.

Re: What are the 'real numbers', really?

#56

Post by igravious »

ColinWright wrote: post #44
igravious wrote: post #30

Earlier quoted context omitted.

I wrote a short paper on the topic once upon a time[1] which you may find interesting. It's part history of math, part philosophy of math. It's not a great paper and most of the insights in it come from others but here is some of the arithmetic of nilpotent[1] infinitesimals as shown in the appendix. Imagine an entity which is not equal to zero but that when raised to the power of 2 or higher is equal to zero! Sounds…

You appear to have an error. You write: (ϵ + 1)(ϵ−1) = −1, or alternately (1 + ϵ)(1 − ϵ) = −1 That alternative should surely be: (1 + ϵ)(1 − ϵ) = 1 Not least, in a commutative system (1+x)(1-x) = 1-x^2. Thus (1 + ϵ)(1 − ϵ) = 1 - ϵ^2 = 1

Thanks, well caught :)

Re: What are the 'real numbers', really?

#57

Post by igravious »

jonsen wrote: post #25
igravious wrote: post #24

Earlier quoted context omitted.

Excuse me. Of course. I was using line and line segment interchangeably there. Which I should have not been doing if I am aiming for clarity but I think my point (ahem) applies to line segments and lines that extend indefinitely in one or two directions. Presumably people will contend that even a line segment "contains" an infinite number of points. But if points have zero extension then even an infinity of them cann…

Would it make a difference if you substituted "infinitesimal extension" for "zero extension"?

That's the thing though. As I understand it, or as it's said to be: points have zero extension. So, no amount of points, not even an infinity of them could ever have extension. But, it should make sense for a line to be composed of entities with infinitesimal extension as you say. I have seen the term linelet used before for these entities.

Charles Sanders Peirce said in 1903, ”Now if we are to accept the common idea of continuity […] we must either say that a continuous line contains no points or […] that the principle of excluded middle does not hold of these points. The principle of excluded middle applies only to an individual […] but places being mere possibilities without actual existence are not individuals.”

Re: What are the 'real numbers', really?

#58

Post by GFK_of_xmaspast »

totemizer wrote: post #32

" It seems that any proper theory of real numbers presupposes some kind of prior theory of algorithms; what they are, how to specify them, how to tell when two of them are the same. Unfortunately there is no such theory." http://njwildberger.wordpress.com/2012/12/02/difficulties-wi...

Guys like that in general have never seemed all that convincing to me.

Re: What are the 'real numbers', really?

#59

Post by velis_vel »

gweinberg wrote: post #46

The problem with "points on a number line" as a definition for real numbers is that it's not clear how you can tell if you have all of them. You can populate a number line as densely as you care to using just rational numbers, but that's not all of them, you're missing out on numbers like the square root of two. You can toss in the non-intergral powers of rational numbers, but you still won't have all of them, you're…

The set of numbers that can be uniquely defined in the English language in a finite number of letters is a countable set, because the set of finite sequences of English letters is countable.

Re: What are the 'real numbers', really?

#60

Post by velis_vel »

igravious wrote: post #57
jonsen wrote: post #25

Earlier quoted context omitted.

Would it make a difference if you substituted "infinitesimal extension" for "zero extension"?

That's the thing though. As I understand it, or as it's said to be: points have zero extension. So, no amount of points, not even an infinity of them could ever have extension. But, it should make sense for a line to be composed of entities with infinitesimal extension as you say. I have seen the term linelet used before for these entities. Charles Sanders Peirce said in 1903, ”Now if we are to accept the common idea…

> That's the thing though. As I understand it, or as it's said to be: points have zero extension. So, no amount of points, not even an infinity of them could ever have extension.

This isn't true for an uncountably infinite set of points, assuming by 'extension' you mean what is usually called 'meausre' in modern mathematics. Modern theory is perfectly fine with saying that a line of nonzero length contains an infinite number of points of zero length, and trying to draw on Euclidean notions definitions of 'point' and 'line' to find conclusions about real analysis is going to be unhelpful.

I'm not sure what that Peirce quote is trying to say.

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