points in a line

Higher-dimensional geometry (previously "Polyshapes").

points in a line

Postby papernuke » Sun Jun 29, 2008 3:36 am

In a line, (as described in my geometry book) there are an infinite amount of points. It also says that points have no length, or thickness, or any dimensions at all (0D).
it says points only have a location, no size.
If that is true, then even if there were an infinite amount of points in a line, then wouldn't the line still be nothing?
because if there were an infinite amount of "locations" on a line, but with no size, then how would the sizeless "things" be able to create a line?
another wording of the same question is
if a line requires one dimension, how can zero dimensional "items" make it up?
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Re: points in a line

Postby Keiji » Sun Jun 29, 2008 6:49 pm

Zero times infinity is undefined, so why can't they make it up? ;)
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Re: points in a line

Postby papernuke » Sun Jun 29, 2008 11:09 pm

What do you mean by "why can't they make it up?" (as in make what up?)

and isn't any number multiplied by zero, zero? How could it be undefined?
because infinity is still a number isnt it?
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Re: points in a line

Postby Keiji » Tue Jul 01, 2008 2:07 pm

Anything divided by zero is infinity; therefore infinity multiplied by zero is anything, i.e. undefined.
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Re: points in a line

Postby wendy » Wed Jul 02, 2008 8:41 am

One must always remember that things can be viewed from content or from context.

The content view supposes a line to be made of points. The context view suggests that a point is made of lines.

In the notion that a line is made of countable points, this supposes alpha_0 = C, which is generally regarded false: there are more points on the line, then can ever be constructed.

In view of context, one notes that in 3d, a point requires three equal signs, while a line requires only two. A point, then is an intersection of a line and a plane, since 3 = 2+1. In this regard, a point is more information than a line, and is a deeper construction.

A line, then is not "made of" points, but "contains" points. There are places on the line that have no construction.
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Re: points in a line

Postby papernuke » Wed Jul 02, 2008 6:02 pm

In the notion that a line is made of countable points, this supposes alpha_0 = C, which is generally regarded false: there are more points on the line, then can ever be constructed.

What does alpha_0 = C mean?

In view of context, one notes that in 3d, a point requires three equal signs, while a line requires only two. A point, then is an intersection of a line and a plane, since 3 = 2+1. In this regard, a point is more information than a line, and is a deeper construction.

I thought in 3D a point needed three coordinates, and 2 for 2D, so what do you mean by three/two equal signs are needed?

A line, then is not "made of" points, but "contains" points. There are places on the line that have no construction.

If a line only "contains" points, then what would it be made of? If (from my first post) a line "contains" an infinite amount of then it wouldn't it just be "made up" of points?
a line is certainly not made of anything "physical" (like atoms) because a line only exists in your imagination (yes?).
Also, what do you mean when you say that there are places on a line that have no construction? Wouldn't what you say mean that some parts of the line are simply "not there" or empty?
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Re: points in a line

Postby wendy » Thu Jul 03, 2008 9:15 am

papernuke wrote:What does alpha_0 = C mean?


In the Cantor theory of infinity, alpha_0 (aleph_0), is the set of countables. C is the set of points in a real section. It supposes that aleph_0 = C is that all points are constructable.

papernuke wrote:I thought in 3D a point needed three coordinates, and 2 for 2D, so what do you mean by three/two equal signs are needed?


A plane can be set with a single equal sign: aX+bY+cZ+d = 0. A line requires two equal signs, a point requires three. The point is that just as one can set dimensions from zero, one can also set dimensions from solid.

papernuke wrote:If a line only "contains" points, then what would it be made of? If (from my first post) a line "contains" an infinite amount of then it wouldn't it just be "made up" of points?a line is certainly not made of anything "physical" (like atoms) because a line only exists in your imagination (yes?).


A line is made of latrix: space of 1d. It is the intersection of a latrix and a dividion that makes a point.

papernuke wrote:Also, what do you mean when you say that there are places on a line that have no construction? Wouldn't what you say mean that some parts of the line are simply "not there" or empty?


The spaces that can not be constructed still exist, but there is no path (telos) to them. Therefore if there were a point (teelon: destination), there is no journey there. This is the implication, for example, of the theory of Gödel's incompleteness theory.

Hayate wrote:Zero times infinity is undefined, so why can't they make it up?


Zero times infinity, is not undefined, but indefinite. None the same, one can do 0/0 reliably, if one knows the parameters of the bounce. In essence, infinity, or 1/0 is not so much a specific number, but an semidefinite number. 0/0 works, when the semidefiniteness contains the definiteness of the division. It's pretty important, really: Euclidean geometry relied on 0/0.
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Re: points in a line

Postby Keiji » Sat Jul 05, 2008 12:11 am

Wendy - complicating a layman's questions since 2006. :mrgreen:

I wrote:Zero times infinity is undefined, so why can't they make it up?


was an attempt to explain in simple terms. I know that isn't exactly how it works.
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Re: points in a line

Postby wendy » Sat Jul 05, 2008 12:40 pm

It is not 'undefined' but 'indefinite'. These are two different words.

Consider, first the case where you have a disk, say 1 inch diam, and a point. You can then draw circles whose centres lie in the disk, and the point is on the circumference. This gives a 'dazzle' of lines, which is fairly easy to see the different lines.

Now move this disk to increasingly far locations. When you repeat the same exercise, most of the lines fall together. At some point, it becomes difficult to tell by construction that the lines are different. You can argue that they are or are not, but the thing is that the disk is in effect, a point on the horizon, and that the circles are replicas of a circle representing the straight line through the local point.

Most of the time, it does not matter which of this dazzle of lines you call "the line". In this way, we have an "indefinite" 1/0 giving a concrete 0.

Sometimes it does matter. You can't assume because this dazzle of lines have centres that lie in a "point at infinity", that this point is like a local point. It is a whole region that behaves as if it _were_ a point.

On the other hand, one can see that one can pass cycles (like remainders) through regions of deep infinity, and still rely, for example, that the selected point is at the crest of the cycle or whatever. For example, the process of replacing x -> 10^x repeatedly, will give a constant remainder in any given power of m^n. So, for example, making a tower of 10's in power, will give ever increasing values, but the _remainder_ when divided by M, however large, and to whatever power, will be constant. So this is a definite value.
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Re: points in a line

Postby Eric B » Thu Oct 23, 2008 12:37 am

Hi all; I use the concept of lines consisting of ∞ points as the basis of my idea of the circle as an ∞-gon (and the sphere as ∞-hedron, and the hypersphere as ∞-choron, and so on).

I've added the stuff I had discussed on this to my own page:
http://members.aol.com/bdmnqr2/essays.html#math (AOL's ftp space is shutting down in a couple of weeks, so this will have to move).

I argue that the so-called "apeirotope" is but an infinitessimal magnification of one of the "points" of a "closed ∞-tope" with a fixed radius.

We end up with a paradox; because the circle, and the endless straight line [apeirogon] are supposed to be the same object! What has happened, is that the straight line, in which the lengths of the sides remained fixed [as we increased the number of sides from 3 to infinity] is an infinitessimal projection of the perimeter of the circle! The sides in the "circle" projection had shrunk down to zero in length! This makes sense, as every line segment is considered to be composed of an infinite number of points, which are zero in length. It has to be, as mathematically, there is no limit to division of length. Take any object with a width, or area, those lengths can always be halved, tenthed, hundredth, etc. Take any space between objects, and it can always be halved, etc. While the lines segments are "next to each other" (adjacent) at 180°; the individual points on the circle are not "next to each other". If you take one point, of zero length, and place it "next to" another point, also of zero length, they won't sit beside one another; but occupy the same spot! (Unless you have some amount of space between them). It would take an infinite number of them to reach the "next" place; and even that is undefinable, as they do not fill any space for there to be any "next place". "Next to" nothing is still nothing!
But since we specify either a radius for a circle, or a length for a line segmant, on our scale, this then covers an infinite number of these infinite points making up the line, which are viewed all at once).
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Re: points in a line

Postby wendy » Sat Oct 25, 2008 9:50 am

There is no paradox: an endless straight line is never a circle, but a horogon. The paradoxes are caused in part by using the same terms for different things, and by making assumptions about completion.

An 'apeirotope' is a tiling. The greek word 'apeiron' is used to express vastness of space 'without (a-) + limit (peri). In this sense, one might cover a plane with hexagons so that there is no limit (edges exposed in the plane). That is, an 'apeirohedron' is 2d patches limitlessly in the same 2d space. All polytopes exist as an apeirotope in _some_ space. Likewise, every apeirotope can become a polytope with a margin-angle > 180° by reducing the curvature of space.

A 'planotope' is a polytope bounded by all faces in the same plane. A planohedron {6,3} would be a tiling of hexagons, and "half-space", eg the tiles on the floor and all the ground below it.

An "infinitope" has infinite faces. However, this does not guarrantee that it is a tiling. One can make an infinitope by marking off the successive radians, ie a {2pi}-gon.

A "horotope" is a polytope whose surface has euclidean geometry, as might arise from a centre or surface at the horizon. In hyperbolic space, this gives a polygon with a margin-angle less than 180°

You can of course take a set of numbers, such as B10 (the set Z, C/10; that is, if x is in the set, so is x/10), which has the feature that between any point in the set, and any other point, there lies additional members. However, the set B10 (as with most sets with constructions), is 'discrete'. That means, that there are points that are _never_ part of the set, regardless of how heavy you pepper the line. The point 1/3 or pi, are not exact decimals, and the line passing through the point 1/3 will pass through the line peppered with B10, at a clear spot.

The complete horizon, when scaled to a finite size, does not give a circle. It remains a horotope, with euclidean geometry. What happens is that when the horizon exists as a real circle H, every circle on the plane has the same radius: one gets the Möbius geometry hight 'inversive geometry'. The polygon {w4}, or horogon, or euclidean line divided equally, can _only_ be inscribed in a horocycle.

In hyperbolic geometry, taking polygons with ever-larger number of edges, does indeed approximate the horogon, and one is scarsely to tell whether the polygon at hand is a {1e80} or {w4}, since the two so exactly follow each other. On the other hand, were one taking a circle as {w4}, the horogon, such that it is not apparent whether x=2x, for example, then every circle is exactly the same size (or rather one can not distinguish the sources of the images as different), and one notes that inversive, not euclidean metric geometry applies to the model.
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Re: points in a line

Postby quickfur » Sat Oct 25, 2008 4:17 pm

One ought to be careful when speaking of infinity, remembering not to treat it as a "number". There are systems under which there are infinite numbers, but these have different infinities which should not be confused with each other. For example, one can "close" the real line by adding a point at infinity, but this should not be conflated with the concept of unbounded iteration (e.g. the number of edges in an apeirogon) because they are two different things.

One system under which one could deal with unbounded iteration is Cantor's system of ordinals, in which case we might say the ordinal omega, the set containing all natural numbers, represents the "number" of edges in an apeirogon. But under this system, one cannot subtract or divide, so it does not make sense to speak of "infinite magnification" or infinitesimals. One should also note that the number of points in a line is much larger than the number of edges in an apeirogon, even though both are infinite. These are two different infinities that cannot be conflated without getting into trouble.

There is a system in which ordinals can be "divided": Conway's surreal numbers. In this system, one can apply such operations as square roots, reciprocation, exponentiation, etc., to arbitrary surreals, which includes all the ordinals. The problem with using this system is that it is large: very large, in fact, so large that it cannot be considered a set under standard set theory. There are so many elements in this system that one cannot, for example, map the results of operations such as dividing by omega, back to the real numbers without causing problems (it would have to collapse a huge infinitude of elements into one, which means it is a one-way map) - one cannot simply apply these operations and get results that even remotely resembles the real number system.

Anyway, the upshot of all this is that one should be aware that "infinity" is a generic term covering a large variety of incompatible concepts, and one cannot conflate them without getting into a contradiction.
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Re: points in a line

Postby Nick » Mon Oct 27, 2008 10:46 pm

I thought infinity x 0 = anything you want, since neither 0 nor infinity are numbers, but concepts.
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Re: points in a line

Postby quickfur » Mon Oct 27, 2008 11:25 pm

Nick wrote:I thought infinity x 0 = anything you want, since neither 0 nor infinity are numbers, but concepts.

Well, every number is a concept. :) There is no physical object which may be called a number, although we can certainly count physical objects and assign a number to them (the number of objects we counted).

Zero is a number, because we can consistently perform arithmetic on it just as we can with the other numbers. Infinity, however, is not (and cannot be) a single number, because attempting to treat it as one causes all kinds of contradictions. There are ways of talking about infinite numbers that won't cause contradictions, but these require much care in order not to get into trouble by assuming they behave a certain way like finite numbers, when they don't.

The thing is that when you go from the finite to the infinite, certain properties that are true in the finite realm are no longer true in the infinite. Furthermore, different properties that are embodied equally in the finite numbers diverge into different kinds of infinity that cannot be reconciled. So we can no longer treat certain things equivalently as we can when the numbers are finite.

For example, in the finite realm, counting numbers (first, second, third, ... etc) are the same as the magnitude numbers (how many objects there are in a collection). In the infinite realm, however, this is not true! Depending on the order you count things, you can end up with different infinite numbers, and they are often not the same as the total number of things you counted.

Also, in the finite realm, you can divide a line segment of finite length into a finite number of shorter line segments, and the sum of the lengths of each segment will add up to the length of the original segment. In the infinite realm, however, this is not always true! You can cut up an infinite length (say, the X-axis) into different infinite numbers of line segments, and their lengths may or may not add up to the original length. For example, you can cut up the X-axis into segments of one unit each, and you will get a set with aleph_0 segments. But you can also cut it up into segments of two units each, and you still get a set with the same number of segments! Then, if you remove one of the segments from the set, you still have aleph_0 segments, but now they no longer add up to the entire X-axis.

Furthermore, you can cut up the line into an infinite collection of points, in which case you get a set with c elements, and c is much, much, much, much larger than aleph_0, even though both are infinite. Note that these are two different infinite numbers, and cannot possibly be treated as the same quantity without getting a contradiction. In fact, no matter how long your original line was, if you cut it up into points you will always get c points. The number of points in a line of length 1 is the same as the number of points in the entire X-axis. In fact, you can take the points that make up a line segment of length 1, and just by rearranging them, you can get line segments of length 2 or length 3, or even infinite length, as well as everything in between. In fact, if you rearranged the points in the X-axis in a suitable way, you can get the entire 2D plane, and if you rearranged them another way, you can get the entire 3D space (or even 4D space, or any finite dimension, or even infinite-dimensional space---for certain values of "infinite"---I told you there are different kinds of infinity).

As you can see, infinite quantities behave very differently from finite quantities, and so one cannot careless talk about them as if they behaved pretty much the same as finite quantities. Otherwise, you easily fall into contradictions and inconsistencies.
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Re: points in a line

Postby Eric B » Thu Oct 30, 2008 12:41 am

wendy wrote:There is no paradox: an endless straight line is never a circle, but a horogon. The paradoxes are caused in part by using the same terms for different things, and by making assumptions about completion.

An 'apeirotope' is a tiling. The greek word 'apeiron' is used to express vastness of space 'without (a-) + limit (peri). In this sense, one might cover a plane with hexagons so that there is no limit (edges exposed in the plane). That is, an 'apeirohedron' is 2d patches limitlessly in the same 2d space. All polytopes exist as an apeirotope in _some_ space. Likewise, every apeirotope can become a polytope with a margin-angle > 180° by reducing the curvature of space.

In hyperbolic geometry, taking polygons with ever-larger number of edges, does indeed approximate the horogon, and one is scarsely to tell whether the polygon at hand is a {1e80} or {w4}, since the two so exactly follow each other. On the other hand, were one taking a circle as {w4}, the horogon, such that it is not apparent whether x=2x, for example, then every circle is exactly the same size (or rather one can not distinguish the sources of the images as different), and one notes that inversive, not euclidean metric geometry applies to the model.

Well, I did not exactly say an endless straight line was a circle. Just that it was an infinitessimal magnification of one of the points of a circle, and hypothetically speaking.

Now, if I understand this last response correctly; then the number of line segments in an apeirogon is really "aleph 0", which is one type of "infinite number", while the number of points in a line segment is another type of infinite number called "c". I had never heard of aleph 0, and I also don't know whether this "c" is a common notation for this (like it is for the speed of light), or whether you just used it for this example.

If this is correct, then that is probably the "paradox" I mentioned. Fixing the length of line segments, and then simply adding them forever until they form an endless line yields one type of infinite number, and fixing the radius instead and letting the sides shrink down to nullitope points making up a circle is another kind of infinite number. If this is true, then I can add that to the essay. But to me the difference was simply one of infinite vs infinitessimal. Or perhaps the notations ∞ vs 1/0; which in one sense is supposed to be the same thing; but then not really, as 1/0 is called "undefinable". But then ∞ is kind of undefinable anyway. Hence, the warnings here about treating it as a regular number. I guess in one of those notations, you're starting out with "infinity'" or the undefinable value of ∞ to represent infinity; while in the second case, you're using definable, finite integers (0, 1) to try to define the same infinite value.

Speaking of horogons and such, what do you all think of the new number line I come up with on the page?
Starting at 0, you go forward with the positive integers as normal (1, 2, 3, etc). But going backwards, instead of using a "negative' mirror image of thepositive integers, you simply use normal substraction and take it backwards. So ...0000000 - 1 would be ...99999999. Sort of the opposite of decimal fractions such as 1/3 (.33333....; which multiplied by 3 is technically .999999... which is basically treated as equalling 1), since there are an infinite number of places to the left of the decimal point, containing zeroes, then if if you subtract 1, make it 9, and then just keep taking one from the place to the next, you end up with an infinite sequence of 9's to the left of the decimal place. You can subtract any finite number from it as normal. subtracting 1 would get ...9999998. subtracting 10 would be ...999990. subtracting 100 would be ...9999900. And so on. Makes subtracting into the negative range easier, but multiplication becomes impossible. ...9999. looks like it is divisible by 9, but since 9 goes into 0, ...99999 is still -1, and primary!

The purpose of this was sort of to try to close up the number line, and make 0 and ∞ meet at the same place. To treat 0 as a power of ten following a number consisting of a sequence of 9's. Of course, the line is still infinitely "open" on both ends. So it becomes what; a parabola, I guess? It just eliminates +∞ and -∞ pretending they are both at the same point, (in our reach) and that the line is really an infinitely large circle.

I tried to search for such a concept, as I would find it hard to believe that I would be the first person to ever ponder such a concept. I guess it might be under a name, which I do not know to search under. Has anyone else ever thought of this?
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Re: points in a line

Postby wendy » Thu Oct 30, 2008 9:49 am

Let's see. The process of creating a polygon rUo (r = edge of adjacent points, U = polygon of short-chord 4 = centred at infinity, o = not used), can give a circle or a straight line. Scaling infinity to this scale supposes that all circles (even straight lines) are identical in size: the relevance gives a geometry called 'inversive geometry'. None the less, representing the horizon as a circle in Euclidean geometry gives misleading results about the nature of infinity.

Aleph_0 is the Cantor-theory model of 'countable infinities'. There are things like Aleph_1 and Aleph_2 as well. C (upper-case), is the number representing the number of points in a line. Whether C=aleph_0 or whatever is an open proposition in this theory. None the less, the theory is heavily dependent on the notion of being able to complete infinity, which as it leads to paradoxes, is itself questionable.

One should regard infinity as a kind of 'instrument variable'. For example, a telescope sees further than you, so its limit of sight is higher. Cars and planes travel faster than walking, and so a long journey is more miles by the faster mode. A calculator that holds 8 digits can only hold logrithms as high as 100,000,000 places. That is, for an 8-digit calculator, we can only see that it has x digits. We can add any sort of number with less than x digits without disturbing the display. A twelve-place calculator would show four places of mantissa and 8 exponent digits here, but would run to the same problem with a 12-digit exponent. There are numbers whose exponents are 12 digits long, which means that we can scarsely tell the difference between x and x^2 with this device.

Yet, some numbers have constructions, like 256^256^256^256. Despite having no means to resolve this number directly, we have still the means to calculate its remainder divided by 17 or 23 or 1093.

An expression like 1/3 = 0.3333333... is only an approximation. In the set B10, 3 is not a unit, and the value 0.333333 is ever an approximation to 1/3. When you multiply an approximation by 3, you are still getting an approximation, so 1 = 0.999999999 is only an approximation. Conversely, 2 and 5 are units of B10, so 1/5 = 0.2, gives 5*1/5 = 1 exactly.

The selection of very large numbers is really a matter of choice: why did you choose 100? It's nothing special, you could have chosen, eg 120 (5!, the first multiple of six surrounded by composites), or even some skew distribution (eg the product of primes with less than 32 primitive roots). Personally, i use two different bases for normal calculations, and the idea that 'desimal is king' is not my cup of tea.

The idea of representing infinigon as a circle is quite old, but it leads to incorrect views of infinity, and is usually used by way of approximating pi.
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Re: points in a line

Postby Eric B » Thu Oct 30, 2008 1:33 pm

Oh, I had never heard anyone else use a circle as an ∞-gon (except implying it when raising m and n in a duoprism to ∞ to yield a duocylinder, and even when I pointed that out in light of my "circle as ∞-gon" idea, it was still not really "bought").

It seems C would be uncountable, because what you would be "counting" would be the infinitessimal "points", which are not countable at all. as I say on my page, you can not even place one "next to" another (without sopace inbetween), as they occupy no space.
So yes, "completing infinity" is what raises the paradox I'm discussing, because the only way to have infinity as a containable whole (as a circle is), is to reduce all its "units" down to zero!
So the countable aleph 0 becomes an uncountable C number of nullitopes.

"instrument variable"; that soulds like a good descriptive term.
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Re: points in a line

Postby quickfur » Fri Oct 31, 2008 5:14 pm

Eric B wrote:Oh, I had never heard anyone else use a circle as an ∞-gon (except implying it when raising m and n in a duoprism to ∞ to yield a duocylinder, and even when I pointed that out in light of my "circle as ∞-gon" idea, it was still not really "bought").

Your idea of "closing" the real line at infinity is interesting, though: and you're not the first one who thought of it. Mathematicians in the past, like Riemann, have thought of "closing the circle", so to speak, by adding a "point at infinity". The term for this is "compactification": you take the set of all real numbers and add one more element to it, called infinity (for this to work out correctly, you cannot distinguish between positive/negative infinity). This then gives the real line some nice closure properties, although one must still be careful not to confuse this infinity with other things commonly thought of as infinity.

(Technically speaking, however, compactification only works out "nicely" if you're dealing with complex numbers on the complex plane, in which case adding the point at infinity yields what is called the Riemann sphere. I don't know enough about compactification to know what are the caveats of using only the real numbers in this case.)

It seems C would be uncountable, because what you would be "counting" would be the infinitessimal "points", which are not countable at all. as I say on my page, you can not even place one "next to" another (without sopace inbetween), as they occupy no space.
So yes, "completing infinity" is what raises the paradox I'm discussing, because the only way to have infinity as a containable whole (as a circle is), is to reduce all its "units" down to zero!
So the countable aleph 0 becomes an uncountable C number of nullitopes.

"instrument variable"; that soulds like a good descriptive term.

One must be careful with intuitive notions when dealing with infinity, because infinite quantities have a lot of unintuitive properties.

Take for example the set of rational numbers: how much "space" does each rational number occupy? If you lay them out in the usual order along the number line, then between any two of them there is always another rational. So they must also occupy "no space", since if they did occupy non-zero space, then you could find a pair of rational numbers that do not have another rational in between (this is impossible, since adding them and dividing by 2 always gives another rational that's in between them). The mathematical term for this property is density: we say that the rational numbers are dense because between every two rationals, no matter how close they may be, there's always another rational.

However, there is a way to rearrange the rationals such that they map 1-to-1 with the natural numbers. That is to say, the number of rationals is the same as the number of natural numbers! This number is aleph_0, and is far less than c, the number of points in the real line. (The notation c is standard notation in set theory.) How can this be true? Simple: write every rational as a quotient of two integers, say p/q. Then make a table, with the rows representing p, and the columns representing q. Every rational then maps to a unique entry in the table (although some table entries may correspond with more than one rational). Now, flatten the table as follows: (1) the first element in the list is the entry at (1,1); (2) the next two entries are (1,2) and (2,1); (3) the next 3 entries are (1,3), (2,2), (3,1), ... and so forth. In other words, we traverse the table via its diagonals. You can see that this will eventually reach any rational number you can think of. Now, lay these diagonals end-to-end, and you have a countable sequence of all rational numbers (although it is not in ascending or descending order of the value of the rationals).

So you see, even though the rationals are dense (in the usual ordering), that does not mean they are uncountable!

With the real numbers, however, it is not possible to rearrange them into a countable sequence, no matter how hard you try. This is not just because we don't know how to, but more because there cannot be any such sequence, because if there were, the number of reals would be the same as the number of naturals, which cannot be true (Cantor proved that there would be a contradiction). In other words, there are so many reals that they simply cannot be "unpacked" into a countable sequence, unlike the rationals.

So the number of rationals (which is equal to the number of naturals) is one kind of infinity, and the number of reals is another kind of infinity, and the first is strictly less than the second, so they are two different infinite quantities. These two infinite quantities are not the same as the "point at infinity" in the Riemann sphere.

In other words, once we get out of the realm of the finite into the realm of the infinite, we realize that there are different kinds of infinity, and they may be related to each other (the number of naturals is less than the number of reals), or they may be completely unrelated to each other (there is no meaningful way to compare the "point at infinity" with, say, the number of reals). The counting numbers are not the same as the cardinal (size) numbers once you are past the finite realm; even though they are related, they are distinct concepts that cannot be conflated.

Coming back to the idea of making a circle out of an infinite number of segments: you should be aware that in taking the limit of polygons with an increasing number of vertices, a "quantum leap" is needed to actually reach the limiting figure, the circle. We should not confuse the act of constructing better polygonal approximations to the circle to the actual attainment of the circle. One way to illustrate this is to consider how we may approximate the set of real numbers by representing them as N-digit approximations. When we use 1 digit, we only have 10 numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10). When we use 2 digits, we have 0.1, 0.2, 0.3, ... 1.0, 1.1, 1.2, ... 9.8, 9.9: we can form 100 numbers. If we add another digit to it, we can form 1000 numbers. We can keep going, and each digit we add multiplies the size of our set by 10. Now, look at the sequence of our set sizes as we increase the number of digits: 10, 100, 1000, 10000, ... etc.. What is the limit of this sequence? The limit is "infinity"... or, to be precise, it's a countable infinity. That is to say, aleph_0 is the limit of the size of our approximating sets. However, if we now make the quantum leap to numbers with an infinite number of digits, suddenly the size of our set is bigger than aleph_0: it is in fact C, because we have now constructed the real numbers. But how can this be? We know that the limit of the sizes of our approximating set is only aleph_0. So how can it be that the set we get by allowing an infinite number of digits suddenly becomes bigger than aleph_0?

The answer is that when we made the quantum leap from a finite number of digits to an infinite number of digits, we went a LOT farther than we thought we did. When we were in the finite realm, each digit we added increased our set by a factor of 10; however, when we jumped from the finite to the infinite, we actually increased the size of our set by C itself. No matter how many finite digits we added, we could only get a countable set, and each time we add more digits, we only added a countable number of numbers to the set. However, an uncountable number of numbers were added in that last step when we jumped from finite numbers of digits to an infinite number of digits. Note well that it is in that last step where an uncountable number of elements were added. This means that what we see in the finite realm when we were merely adding a finite number of digits does not even begin to approximate what happens in that last step. Jumping from the finite to the infinite is a leap so huge that we could not have anticipated the uncountable number of elements that would be added.

Coming back to approximating the circle: we can make polygons of as many (finite) vertices as we want, and the result is still a countable number of vertices, with a countable number of line segments. But in that last step, when we finally say, OK, let's jump from a finite number of vertices to an infinite number of vertices, something happens: we made that leap from the finite to the infinite, and in the process, an uncountable number of "sides" were added, resulting in a circle.

In other words, we should not imagine that we can somehow "magnify" a circle by an "infinite" factor (whatever that means) so that we can see individual line segments; actually, no matter how much you magnify a circle, there will still be the same number of points: an uncountable number of points between every two points. Even though the finite polygonal approximations of the circle could be magnified so that we can see the edges, when we made that leap out of the finite realm into the infinite realm something happened: there are no longer any line segments left; they have been replaced by an uncountable number of points. Even though we expected there to be only aleph_0 number of segments, since aleph_0 is the upper limit of the number of segments in our finite approximations, just as our sets of numbers approximating the reals earlier suddenly acquired an uncountable number of elements in the last step (the leap into infinity), so our polygonal approximations suddenly acquire an uncountable number of "vertices" in the last step, when we leaped into infinity.

The bottom line is, our intuition about how things behave in the finite realm does not generalize to the infinite realm. We thought we were only approaching aleph_0, but actually when we made the leap, we ended up a lot farther than aleph_0; we ended up in C.
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Re: points in a line

Postby Eric B » Mon Nov 03, 2008 3:09 am

quickfur wrote:
Eric B wrote:Oh, I had never heard anyone else use a circle as an ∞-gon (except implying it when raising m and n in a duoprism to ∞ to yield a duocylinder, and even when I pointed that out in light of my "circle as ∞-gon" idea, it was still not really "bought").

Your idea of "closing" the real line at infinity is interesting, though: and you're not the first one who thought of it. Mathematicians in the past, like Riemann, have thought of "closing the circle", so to speak, by adding a "point at infinity". The term for this is "compactification": you take the set of all real numbers and add one more element to it, called infinity (for this to work out correctly, you cannot distinguish between positive/negative infinity). This then gives the real line some nice closure properties, although one must still be careful not to confuse this infinity with other things commonly thought of as infinity.

(Technically speaking, however, compactification only works out "nicely" if you're dealing with complex numbers on the complex plane, in which case adding the point at infinity yields what is called the Riemann sphere. I don't know enough about compactification to know what are the caveats of using only the real numbers in this case.)
Still, did any of these other mathematicians ever speak of extending the positive number line backwards with "...999999" (representing -1)?

It seems C would be uncountable, because what you would be "counting" would be the infinitessimal "points", which are not countable at all. as I say on my page, you can not even place one "next to" another (without sopace inbetween), as they occupy no space.
So yes, "completing infinity" is what raises the paradox I'm discussing, because the only way to have infinity as a containable whole (as a circle is), is to reduce all its "units" down to zero!
So the countable aleph 0 becomes an uncountable C number of nullitopes.

"instrument variable"; that soulds like a good descriptive term.

One must be careful with intuitive notions when dealing with infinity, because infinite quantities have a lot of unintuitive properties.

Take for example the set of rational numbers: how much "space" does each rational number occupy? If you lay them out in the usual order along the number line, then between any two of them there is always another rational. So they must also occupy "no space", since if they did occupy non-zero space, then you could find a pair of rational numbers that do not have another rational in between (this is impossible, since adding them and dividing by 2 always gives another rational that's in between them). The mathematical term for this property is density: we say that the rational numbers are dense because between every two rationals, no matter how close they may be, there's always another rational.
This was something I was covering in my essay.

However, there is a way to rearrange the rationals such that they map 1-to-1 with the natural numbers. That is to say, the number of rationals is the same as the number of natural numbers! This number is aleph_0, and is far less than c, the number of points in the real line. (The notation c is standard notation in set theory.) How can this be true? Simple: write every rational as a quotient of two integers, say p/q. Then make a table, with the rows representing p, and the columns representing q. Every rational then maps to a unique entry in the table (although some table entries may correspond with more than one rational). Now, flatten the table as follows: (1) the first element in the list is the entry at (1,1); (2) the next two entries are (1,2) and (2,1); (3) the next 3 entries are (1,3), (2,2), (3,1), ... and so forth. In other words, we traverse the table via its diagonals. You can see that this will eventually reach any rational number you can think of. Now, lay these diagonals end-to-end, and you have a countable sequence of all rational numbers (although it is not in ascending or descending order of the value of the rationals).

So you see, even though the rationals are dense (in the usual ordering), that does not mean they are uncountable!

With the real numbers, however, it is not possible to rearrange them into a countable sequence, no matter how hard you try. This is not just because we don't know how to, but more because there cannot be any such sequence, because if there were, the number of reals would be the same as the number of naturals, which cannot be true (Cantor proved that there would be a contradiction). In other words, there are so many reals that they simply cannot be "unpacked" into a countable sequence, unlike the rationals.
I misunderstood the term "countable". I'm still trying to understand what "uncountable" really is, then. Wikipedia mentioned injective functions.

So the number of rationals (which is equal to the number of naturals) is one kind of infinity, and the number of reals is another kind of infinity, and the first is strictly less than the second, so they are two different infinite quantities. These two infinite quantities are not the same as the "point at infinity" in the Riemann sphere.

In other words, once we get out of the realm of the finite into the realm of the infinite, we realize that there are different kinds of infinity, and they may be related to each other (the number of naturals is less than the number of reals), or they may be completely unrelated to each other (there is no meaningful way to compare the "point at infinity" with, say, the number of reals). The counting numbers are not the same as the cardinal (size) numbers once you are past the finite realm; even though they are related, they are distinct concepts that cannot be conflated.

Coming back to the idea of making a circle out of an infinite number of segments: you should be aware that in taking the limit of polygons with an increasing number of vertices, a "quantum leap" is needed to actually reach the limiting figure, the circle. We should not confuse the act of constructing better polygonal approximations to the circle to the actual attainment of the circle. One way to illustrate this is to consider how we may approximate the set of real numbers by representing them as N-digit approximations. When we use 1 digit, we only have 10 numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10). When we use 2 digits, we have 0.1, 0.2, 0.3, ... 1.0, 1.1, 1.2, ... 9.8, 9.9: we can form 100 numbers. If we add another digit to it, we can form 1000 numbers. We can keep going, and each digit we add multiplies the size of our set by 10. Now, look at the sequence of our set sizes as we increase the number of digits: 10, 100, 1000, 10000, ... etc.. What is the limit of this sequence? The limit is "infinity"... or, to be precise, it's a countable infinity. That is to say, aleph_0 is the limit of the size of our approximating sets. However, if we now make the quantum leap to numbers with an infinite number of digits, suddenly the size of our set is bigger than aleph_0: it is in fact C, because we have now constructed the real numbers. But how can this be? We know that the limit of the sizes of our approximating set is only aleph_0. So how can it be that the set we get by allowing an infinite number of digits suddenly becomes bigger than aleph_0?

The answer is that when we made the quantum leap from a finite number of digits to an infinite number of digits, we went a LOT farther than we thought we did. When we were in the finite realm, each digit we added increased our set by a factor of 10; however, when we jumped from the finite to the infinite, we actually increased the size of our set by C itself. No matter how many finite digits we added, we could only get a countable set, and each time we add more digits, we only added a countable number of numbers to the set. However, an uncountable number of numbers were added in that last step when we jumped from finite numbers of digits to an infinite number of digits. Note well that it is in that last step where an uncountable number of elements were added. This means that what we see in the finite realm when we were merely adding a finite number of digits does not even begin to approximate what happens in that last step. Jumping from the finite to the infinite is a leap so huge that we could not have anticipated the uncountable number of elements that would be added.

Coming back to approximating the circle: we can make polygons of as many (finite) vertices as we want, and the result is still a countable number of vertices, with a countable number of line segments. But in that last step, when we finally say, OK, let's jump from a finite number of vertices to an infinite number of vertices, something happens: we made that leap from the finite to the infinite, and in the process, an uncountable number of "sides" were added, resulting in a circle.

In other words, we should not imagine that we can somehow "magnify" a circle by an "infinite" factor (whatever that means) so that we can see individual line segments; actually, no matter how much you magnify a circle, there will still be the same number of points: an uncountable number of points between every two points. Even though the finite polygonal approximations of the circle could be magnified so that we can see the edges, when we made that leap out of the finite realm into the infinite realm something happened: there are no longer any line segments left; they have been replaced by an uncountable number of points. Even though we expected there to be only aleph_0 number of segments, since aleph_0 is the upper limit of the number of segments in our finite approximations, just as our sets of numbers approximating the reals earlier suddenly acquired an uncountable number of elements in the last step (the leap into infinity), so our polygonal approximations suddenly acquire an uncountable number of "vertices" in the last step, when we leaped into infinity.

The bottom line is, our intuition about how things behave in the finite realm does not generalize to the infinite realm. We thought we were only approaching aleph_0, but actually when we made the leap, we ended up a lot farther than aleph_0; we ended up in C.
Basically, the "leap" is what I was discussing in the essay. (I've now set up the new webspace, purchasing my own domain name. It's at http://www.erictb.info/essays.html#math)
So aleph 0 is the "approachable" infinity, and C is the hypothetical "reached" infinity. Of course, you can't reach infinity (either aleph or C), so with aleph, you just keep approaching it forever, and with c, you make the leap; but then, you're no longer really on the "line" of numbers. So the segments become replaced with the points. So when I say "magnification"; it's not actual magnification, of course, because, it's infinitessimal magnification; and what is that, really? It itself is the quantum leap from the finite realm to the infinite realm.
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Re: points in a line

Postby quickfur » Mon Nov 03, 2008 7:28 pm

Eric B wrote:
quickfur wrote:
Eric B wrote:Oh, I had never heard anyone else use a circle as an ∞-gon (except implying it when raising m and n in a duoprism to ∞ to yield a duocylinder, and even when I pointed that out in light of my "circle as ∞-gon" idea, it was still not really "bought").

Your idea of "closing" the real line at infinity is interesting, though: and you're not the first one who thought of it. Mathematicians in the past, like Riemann, have thought of "closing the circle", so to speak, by adding a "point at infinity". The term for this is "compactification": you take the set of all real numbers and add one more element to it, called infinity (for this to work out correctly, you cannot distinguish between positive/negative infinity). This then gives the real line some nice closure properties, although one must still be careful not to confuse this infinity with other things commonly thought of as infinity.

(Technically speaking, however, compactification only works out "nicely" if you're dealing with complex numbers on the complex plane, in which case adding the point at infinity yields what is called the Riemann sphere. I don't know enough about compactification to know what are the caveats of using only the real numbers in this case.)
Still, did any of these other mathematicians ever speak of extending the positive number line backwards with "...999999" (representing -1)?

In fact, this has been studied before. Only, the set of such "numbers" have rather strange properties (unfortunately I can't recall them off the top of my head) that makes them difficult to map back to the usual numbers.

[...]
With the real numbers, however, it is not possible to rearrange them into a countable sequence, no matter how hard you try. This is not just because we don't know how to, but more because there cannot be any such sequence, because if there were, the number of reals would be the same as the number of naturals, which cannot be true (Cantor proved that there would be a contradiction). In other words, there are so many reals that they simply cannot be "unpacked" into a countable sequence, unlike the rationals.
I misunderstood the term "countable". I'm still trying to understand what "uncountable" really is, then. Wikipedia mentioned injective functions.

Well, maybe I should explain precisely what "countable" means, first. :)

The motivating intuition is this: think back of when you were a child at a birthday party (or think of some child at a birthday party), and you haven't learnt to count past 5 yet. Suppose more than 5 people were present (say, 10 or so). You have a number of hats (also more than 5) that is to be distributed to each person present. How would you know that the number of hats equals the number of people? In theory, if you could count the number of hats and then count the number of people, you could compare the numbers to see if they were equal; however, the assumption is that you haven't learnt to count that high yet. What's another way of making the comparison that doesn't require you to count that high? Here's one solution: pass out the hats, one to each person, and if at the end, there are no hats left over and there are no persons without a hat, then you know that there are as many hats as there are people. This is basically the "pigeon hole principle" (if the number of pigeons and the number of holes are equal, then there will be no pigeons without a hole, and no hole without a pigeon).

Now, the idea behind Cantor's infinities (countable or otherwise) is this: we haven't learnt to count up to infinity yet (and probably never will!), but we can compare two infinities by establishing injective functions between them. If there exists at least one bijective function between two sets (possibly infinite), then we say that the two sets have equal cardinality (size). Note that in the infinite realm, it is possible for a proper subset of a set to have the same cardinality as itself, so we only require the existence of one bijective function between them to establish equality of cardinality.

As a simple example, consider the set of all natural numbers, and the set of even numbers. It is simple to check that the function f(n) = 2*n is a one-to-one function between these two sets. Hence, by the definition above, the cardinality of the natural numbers equals the cardinality of the even numbers. (This may seem strange at first: the even numbers are only half of the set of natural numbers; how can they still have the same number of elements, so to speak? The answer is that their cardinality is infinite, and therefore has properties that are different from the finite numbers. In fact, under some formal systems, an infinite set is defined to be any set that has the same cardinality as one of its proper subsets.)

The set of integers also have the same cardinality as the natural numbers: take any integer, and if it is positive, multiply it by 2, and if it is negative, multiply it by -2 and then subtract 1. This is a bijection from the integers to the natural numbers.

The set of rationals are also of the same cardinality as the set of natural numbers: using the method of turning the rationals into a linear chain, as I've described in my previous post, we can, after forming that chain, assign them to the natural numbers in order, thereby forming a bijection. Using the same method, we can also prove that the set of vectors with integer coordinates have the same cardinality as the set of natural numbers (first, map the integer coordinates to the natural numbers, as described in the previous paragraph, then form a table indexed by each respective coordinate, then apply the diagonal traversal trick to map each coordinate pair to a unique natural number). By induction, the set of n-dimensional vectors with integer coordinates have the same cardinality as the set of natural numbers, for all finite n.

So far, it seems that our definition of cardinality has produced only a single infinity, the cardinality of the natural numbers. This we call the countable infinity. When we say a set is countable, we mean that it is either finite, or has a bijection with the natural numbers. One consequence of the countable infinity is that if a set is countable, then it can be laid on in a sequence: since it is bijective with the natural numbers, we can arrange them in the order of the natural numbers and get a sequence that eventually covers the entire set. In other words, if it is possible to lay out a set in a (discrete) sequence, then it has the cardinality of the natural numbers. The cardinality of the natural numbers is denoted aleph_0.

However, Cantor proved that there is another infinity which is different from the countable infinity. In fact, it must be strictly greater than the countable infinity, because there is no bijection possible between this infinity and the natural numbers. In other words, this is an uncountable infinity. An uncountable infinity is one that is greater than the cardinality of the natural numbers, and there is no bijection between them. (Note that it is not sufficient to find one non-bijective function between them; one must prove that it is not possible to find any bijective function.) This uncountable infinity is the cardinality of the real numbers. It is not possible to arrange the real numbers in a (discrete) sequence that eventually covers all possible reals (because if we could, they would have the cardinality of the natural numbers).

I won't attempt to prove that the cardinality of the reals must be strictly greater than the cardinality of the naturals; you can find it online in many places. The cardinality of the reals is denoted by c, which stands for "continuum" (as opposed to the countable infinity, which can always be rearranged into a discrete sequence).

[...]Basically, the "leap" is what I was discussing in the essay. (I've now set up the new webspace, purchasing my own domain name. It's at http://www.erictb.info/essays.html#math)
So aleph 0 is the "approachable" infinity, and C is the hypothetical "reached" infinity. Of course, you can't reach infinity (either aleph or C), so with aleph, you just keep approaching it forever, and with c, you make the leap; but then, you're no longer really on the "line" of numbers. So the segments become replaced with the points. So when I say "magnification"; it's not actual magnification, of course, because, it's infinitessimal magnification; and what is that, really? It itself is the quantum leap from the finite realm to the infinite realm.

It helps to be precise when dealing with infinity, because there are many pitfalls that one can get stuck in if one is not careful. I don't think it's accurate to think of a circle as being a polygon where the line segments are "replaced" with points: the circle is the limiting shape of the sequence of polygons with increasing degree; it is not one among them. Just because a sequence approaches a limit, does not mean that the limit shares any of the properties of any of the members of the sequence at all.

For example, the sequence of icosahedra with decreasing radius, with each successive radius being half of the previous one, has the limit at a single point. Does that mean that if you magnify a point "by an infinite amount", you will get an icosahedron? Of course not; the limit of the sequence of cubes with decreasing radius is also a point, so if you magnify a point by an "infinite amount", will you get a cube or an icosahedron? You will get neither, because it is not meaningful to speak of infinite magnification in this manner. The point is not a member of the sequence of shrinking icosahedra, neither is it a member of the sequence of shrinking cubes, nor a member of any sequence of any shrinking figure, for that matter. It is the limit of each of these sequences, but it stands apart from all of them. The process of taking the limit is not invertible, even if it does seem rather compelling sometimes!

In the same way, the circle is a limit of the sequence of polygons with increasing degree, but that sequence is not the only sequence that converges into a circle. One could, for example, consider the sequence of star polygons with Schäfli symbol {n/2}. (I.e., the star polygon with n vertices where each edge connects every other vertex.) When n=1, we get the pentagram; when n=2, we get the hexagram, etc.. See http://en.wikipedia.org/wiki/Image:Regular_Star_Polygons.jpg: our sequence is the one along the line marked "n/2". You can see that as n increases, the non-convex boundary of the star becomes shallower and shallower. What is the limit of this sequence? It should be clear that this sequence also approaches the circle, since at the limit, the non-convex boundary will flatten into a convex boundary. So, given a circle magnified by an "infinite" amount, why wouldn't we see a star polygon segment instead of just a line?

In fact, we might as well take this to its logical conclusion: the sequence of star polygons {n/3} also converges to the circle! It will take "longer" for the non-convex boundary to "flatten out", sure, but "at infinity", so to speak, it does eventually flatten out. So now we have a problem: if it were in fact possible to magnify a circle by an infinite amount, then which of these sequences should be obtained? What makes us think that a sequence of edges is the only possibility? Why wouldn't we also obtain two sequences of edges, each connecting every other vertex (corresponding with the {n/2} stars sequence)? Why shouldn't we also obtain three sequences of edges, each connecting every three vertices (corresponding with the {n/3} stars sequence)? The circle is the limit of all of these sequences, but it is not a member of any of them, and so properties of members of these sequences cannot always be extrapolated to the circle.

From another point of view, the impossibility of magnifying a circle "by an infinite amount" to see individual edges is also indicated by the fact that the number of points on the circle is uncountable, and therefore cannot be arranged in a sequence. (If it could, it would be countable, which is a contradiction.) It is not possible to magnify it so that the points become separated; because by the very definition of the circle being a continuous curve, it must have as many points as the continuum, and it is not possible to magnify the continuum so that its points become separated (since it wouldn't be the continuum otherwise!). In fact, it is not even possible to rearrange the points in the continuum into a sequence, much less separate them while retaining their usual order.
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Re: points in a line

Postby Eric B » Wed Nov 05, 2008 8:13 pm

quickfur wrote:
Eric B wrote: Still, did any of these other mathematicians ever speak of extending the positive number line backwards with "...999999" (representing -1)?

In fact, this has been studied before.
Oh, by who, and where can I find info on it. Is there a name for this branch of math?
Only, the set of such "numbers" have rather strange properties (unfortunately I can't recall them off the top of my head) that makes them difficult to map back to the usual numbers.
This is probably like what I mentioned. Like ...99999. not being divisible by 9, even though it consists of all 9's. And that multiplication and division become meaningless in that range (Because where 0 is the center of symmetry for addition; 1 is the center of symmetry for multiplication).

Well, maybe I should explain precisely what "countable" means, first. :)

The motivating intuition is this: think back of when you were a child at a birthday party (or think of some child at a birthday party), and you haven't learnt to count past 5 yet. Suppose more than 5 people were present (say, 10 or so). You have a number of hats (also more than 5) that is to be distributed to each person present. How would you know that the number of hats equals the number of people? In theory, if you could count the number of hats and then count the number of people, you could compare the numbers to see if they were equal; however, the assumption is that you haven't learnt to count that high yet. What's another way of making the comparison that doesn't require you to count that high? Here's one solution: pass out the hats, one to each person, and if at the end, there are no hats left over and there are no persons without a hat, then you know that there are as many hats as there are people. This is basically the "pigeon hole principle" (if the number of pigeons and the number of holes are equal, then there will be no pigeons without a hole, and no hole without a pigeon).

Now, the idea behind Cantor's infinities (countable or otherwise) is this: we haven't learnt to count up to infinity yet (and probably never will!), but we can compare two infinities by establishing injective functions between them. If there exists at least one bijective function between two sets (possibly infinite), then we say that the two sets have equal cardinality (size). Note that in the infinite realm, it is possible for a proper subset of a set to have the same cardinality as itself, so we only require the existence of one bijective function between them to establish equality of cardinality.

As a simple example, consider the set of all natural numbers, and the set of even numbers. It is simple to check that the function f(n) = 2*n is a one-to-one function between these two sets. Hence, by the definition above, the cardinality of the natural numbers equals the cardinality of the even numbers. (This may seem strange at first: the even numbers are only half of the set of natural numbers; how can they still have the same number of elements, so to speak? The answer is that their cardinality is infinite, and therefore has properties that are different from the finite numbers. In fact, under some formal systems, an infinite set is defined to be any set that has the same cardinality as one of its proper subsets.)

The set of integers also have the same cardinality as the natural numbers: take any integer, and if it is positive, multiply it by 2, and if it is negative, multiply it by -2 and then subtract 1. This is a bijection from the integers to the natural numbers.

The set of rationals are also of the same cardinality as the set of natural numbers: using the method of turning the rationals into a linear chain, as I've described in my previous post, we can, after forming that chain, assign them to the natural numbers in order, thereby forming a bijection. Using the same method, we can also prove that the set of vectors with integer coordinates have the same cardinality as the set of natural numbers (first, map the integer coordinates to the natural numbers, as described in the previous paragraph, then form a table indexed by each respective coordinate, then apply the diagonal traversal trick to map each coordinate pair to a unique natural number). By induction, the set of n-dimensional vectors with integer coordinates have the same cardinality as the set of natural numbers, for all finite n.

So far, it seems that our definition of cardinality has produced only a single infinity, the cardinality of the natural numbers. This we call the countable infinity. When we say a set is countable, we mean that it is either finite, or has a bijection with the natural numbers. One consequence of the countable infinity is that if a set is countable, then it can be laid on in a sequence: since it is bijective with the natural numbers, we can arrange them in the order of the natural numbers and get a sequence that eventually covers the entire set. In other words, if it is possible to lay out a set in a (discrete) sequence, then it has the cardinality of the natural numbers. The cardinality of the natural numbers is denoted aleph_0.

However, Cantor proved that there is another infinity which is different from the countable infinity. In fact, it must be strictly greater than the countable infinity, because there is no bijection possible between this infinity and the natural numbers. In other words, this is an uncountable infinity. An uncountable infinity is one that is greater than the cardinality of the natural numbers, and there is no bijection between them. (Note that it is not sufficient to find one non-bijective function between them; one must prove that it is not possible to find any bijective function.) This uncountable infinity is the cardinality of the real numbers. It is not possible to arrange the real numbers in a (discrete) sequence that eventually covers all possible reals (because if we could, they would have the cardinality of the natural numbers).

I won't attempt to prove that the cardinality of the reals must be strictly greater than the cardinality of the naturals; you can find it online in many places. The cardinality of the reals is denoted by c, which stands for "continuum" (as opposed to the countable infinity, which can always be rearranged into a discrete sequence).
OK; thanks. I thought on the wikipedia info more, and I think I started to get it.

[...]Basically, the "leap" is what I was discussing in the essay. (I've now set up the new webspace, purchasing my own domain name. It's at http://www.erictb.info/essays.html#math)
So aleph 0 is the "approachable" infinity, and C is the hypothetical "reached" infinity. Of course, you can't reach infinity (either aleph or C), so with aleph, you just keep approaching it forever, and with c, you make the leap; but then, you're no longer really on the "line" of numbers. So the segments become replaced with the points. So when I say "magnification"; it's not actual magnification, of course, because, it's infinitessimal magnification; and what is that, really? It itself is the quantum leap from the finite realm to the infinite realm.

It helps to be precise when dealing with infinity, because there are many pitfalls that one can get stuck in if one is not careful. I don't think it's accurate to think of a circle as being a polygon where the line segments are "replaced" with points: the circle is the limiting shape of the sequence of polygons with increasing degree; it is not one among them. Just because a sequence approaches a limit, does not mean that the limit shares any of the properties of any of the members of the sequence at all.
"Replaced" was not meant to be taken literally. I fact, it was you who first used the term to describe what I was saying, and I just picked up from there. I always had said "shrunk". But of course, that term will only carry but so far as well, and faces that same "leap" at some point.

I have been treating it as "one among them" because like them, it is a hull with a radius. It has its higher dimensional analogues like the simplex and orthotope, et al.

For example, the sequence of icosahedra with decreasing radius, with each successive radius being half of the previous one, has the limit at a single point. Does that mean that if you magnify a point "by an infinite amount", you will get an icosahedron? Of course not; the limit of the sequence of cubes with decreasing radius is also a point, so if you magnify a point by an "infinite amount", will you get a cube or an icosahedron? You will get neither, because it is not meaningful to speak of infinite magnification in this manner. The point is not a member of the sequence of shrinking icosahedra, neither is it a member of the sequence of shrinking cubes, nor a member of any sequence of any shrinking figure, for that matter. It is the limit of each of these sequences, but it stands apart from all of them. The process of taking the limit is not invertible, even if it does seem rather compelling sometimes!

In the same way, the circle is a limit of the sequence of polygons with increasing degree, but that sequence is not the only sequence that converges into a circle. One could, for example, consider the sequence of star polygons with Schäfli symbol {n/2}. (I.e., the star polygon with n vertices where each edge connects every other vertex.) When n=1, we get the pentagram; when n=2, we get the hexagram, etc.. See http://en.wikipedia.org/wiki/Image:Regular_Star_Polygons.jpg: our sequence is the one along the line marked "n/2". You can see that as n increases, the non-convex boundary of the star becomes shallower and shallower. What is the limit of this sequence? It should be clear that this sequence also approaches the circle, since at the limit, the non-convex boundary will flatten into a convex boundary. So, given a circle magnified by an "infinite" amount, why wouldn't we see a star polygon segment instead of just a line?

In fact, we might as well take this to its logical conclusion: the sequence of star polygons {n/3} also converges to the circle! It will take "longer" for the non-convex boundary to "flatten out", sure, but "at infinity", so to speak, it does eventually flatten out. So now we have a problem: if it were in fact possible to magnify a circle by an infinite amount, then which of these sequences should be obtained? What makes us think that a sequence of edges is the only possibility? Why wouldn't we also obtain two sequences of edges, each connecting every other vertex (corresponding with the {n/2} stars sequence)? Why shouldn't we also obtain three sequences of edges, each connecting every three vertices (corresponding with the {n/3} stars sequence)? The circle is the limit of all of these sequences, but it is not a member of any of them, and so properties of members of these sequences cannot always be extrapolated to the circle.
I kind of disagree, because what you describe is basically a similar principle to how in 1D, the line segment represents the simplex; orthotope, cross-tope, as well as the hypersphere. (Which since I am considering an ∞-tope; hence, "one among them"; but now; it's reduced to a di-1tope). So what really is a higher dimensional analogue of the line segment? It could be just more line segments; making up the boundaries of polygons, polyhedra, polychora, and so forth. In another sense, it's corresponds to whole polygons, polyhedra, polychora themselves. Or, any of the "families" mentioned. Basically, it corresponds to all of them! The correspondences collapse because of the reduction in dimensions.
So yes, by saying that a point could represent a cube or icosahedron; basically; what you're saying is that in 0D, the nullitope would have to correspond to all polytopes, and that would include the cube and icosahedron. And saying that "if you magnify a point by an 'infinite amount', will you get a cube or an icosahedron?" is basically posing the problem of 0 × ∞ = n. This equation is generally "not allowed"; but hypothetically, any number would possibly fit, because any n/0=∞; and by extension; n/∞=0. So in that sense, a point (representing 0 × ∞) could hypothetically stretch out to anything.

Likewise, with the star analogy, as n approaches ∞, for any the so-called "points" flatten out towards the surface; hence, all finite densities are collapsed onto the perimeter. So yes, the circle would represent all finite Schlafli densities (I also looked into what the "highest" density would be. Of course, it would also be infinite, but it would swell either into an infinitely dense solid disk (If the total diameter is fixed), or swell outward to infinity; still with infinite density (if the radius is fixed). An infinite sequence of polygons rotated at 360/∞ would produce produce an infinitely dense ring, and the symbol would still be {n/∞}.

Of course, allof this is abstract, and not "real" in our everyday sense. So I am quite aware of the limitations.

From another point of view, the impossibility of magnifying a circle "by an infinite amount" to see individual edges is also indicated by the fact that the number of points on the circle is uncountable, and therefore cannot be arranged in a sequence. (If it could, it would be countable, which is a contradiction.) It is not possible to magnify it so that the points become separated; because by the very definition of the circle being a continuous curve, it must have as many points as the continuum, and it is not possible to magnify the continuum so that its points become separated (since it wouldn't be the continuum otherwise!). In fact, it is not even possible to rearrange the points in the continuum into a sequence, much less separate them while retaining their usual order.

Again, this is what I discuss, but the "magnification" is only a hypothetical concept. It is describing the "quantum leap" from aleph 0 to C for lack of any better way of putting it.
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Re: points in a line

Postby quickfur » Wed Nov 05, 2008 11:33 pm

Eric B wrote:
quickfur wrote:
Eric B wrote: Still, did any of these other mathematicians ever speak of extending the positive number line backwards with "...999999" (representing -1)?

In fact, this has been studied before.
Oh, by who, and where can I find info on it. Is there a name for this branch of math?

It's not really a branch of math per se, it was just some random musings by a mathematician. The properties of such "numbers" do not lend themselves to very many useful applications (even within a pure math context), so they aren't really worth studying very much.

Only, the set of such "numbers" have rather strange properties (unfortunately I can't recall them off the top of my head) that makes them difficult to map back to the usual numbers.
This is probably like what I mentioned. Like ...99999. not being divisible by 9, even though it consists of all 9's. And that multiplication and division become meaningless in that range (Because where 0 is the center of symmetry for addition; 1 is the center of symmetry for multiplication).

The usual term is "identity".

As for divisibility, ...99999 is definitely divisible by 9: it divides evenly into ...11111. It should not be surprising that an "infinite" number divides into another "infinite" number. :-)

Well, maybe I should explain precisely what "countable" means, first. :)
[...]
OK; thanks. I thought on the wikipedia info more, and I think I started to get it.

Well, this is what is commonly referred to as Cantorian infinity, or, as Cantor fans would say, "Cantor's paradise". One should keep in mind that this is not the only possible treatment of infinity.

[...]
It helps to be precise when dealing with infinity, because there are many pitfalls that one can get stuck in if one is not careful. I don't think it's accurate to think of a circle as being a polygon where the line segments are "replaced" with points: the circle is the limiting shape of the sequence of polygons with increasing degree; it is not one among them. Just because a sequence approaches a limit, does not mean that the limit shares any of the properties of any of the members of the sequence at all.
"Replaced" was not meant to be taken literally. I fact, it was you who first used the term to describe what I was saying, and I just picked up from there. I always had said "shrunk". But of course, that term will only carry but so far as well, and faces that same "leap" at some point.

Mea culpa. :oops: That'll teach me to reply to something off the top of my head without re-reading the thread. :P

[...]
[...]
For example, the sequence of icosahedra with decreasing radius, with each successive radius being half of the previous one, has the limit at a single point. Does that mean that if you magnify a point "by an infinite amount", you will get an icosahedron? Of course not; the limit of the sequence of cubes with decreasing radius is also a point, so if you magnify a point by an "infinite amount", will you get a cube or an icosahedron? You will get neither, because it is not meaningful to speak of infinite magnification in this manner. The point is not a member of the sequence of shrinking icosahedra, neither is it a member of the sequence of shrinking cubes, nor a member of any sequence of any shrinking figure, for that matter. It is the limit of each of these sequences, but it stands apart from all of them. The process of taking the limit is not invertible, even if it does seem rather compelling sometimes!
[...]
I kind of disagree, because what you describe is basically a similar principle to how in 1D, the line segment represents the simplex; orthotope, cross-tope, as well as the hypersphere. (Which since I am considering an ∞-tope; hence, "one among them"; but now; it's reduced to a di-1tope). So what really is a higher dimensional analogue of the line segment? It could be just more line segments; making up the boundaries of polygons, polyhedra, polychora, and so forth. In another sense, it's corresponds to whole polygons, polyhedra, polychora themselves. Or, any of the "families" mentioned. Basically, it corresponds to all of them! The correspondences collapse because of the reduction in dimensions.
So yes, by saying that a point could represent a cube or icosahedron; basically; what you're saying is that in 0D, the nullitope would have to correspond to all polytopes, and that would include the cube and icosahedron. And saying that "if you magnify a point by an 'infinite amount', will you get a cube or an icosahedron?" is basically posing the problem of 0 × ∞ = n. This equation is generally "not allowed"; but hypothetically, any number would possibly fit, because any n/0=∞; and by extension; n/∞=0. So in that sense, a point (representing 0 × ∞) could hypothetically stretch out to anything.

True.

Likewise, with the star analogy, as n approaches ∞, for any the so-called "points" flatten out towards the surface; hence, all finite densities are collapsed onto the perimeter. So yes, the circle would represent all finite Schlafli densities (I also looked into what the "highest" density would be. Of course, it would also be infinite, but it would swell either into an infinitely dense solid disk (If the total diameter is fixed), or swell outward to infinity; still with infinite density (if the radius is fixed). An infinite sequence of polygons rotated at 360/∞ would produce produce an infinitely dense ring, and the symbol would still be {n/∞}.

Of course, allof this is abstract, and not "real" in our everyday sense. So I am quite aware of the limitations.

Not sure what you mean by "rotated at 360/∞"---presumably an infinitesimal rotation that eventually spans all possible rotations about the origin. Again, this is where one has to be careful about generalizing finite concepts into infinity. Saying that an infinite number of rotations about 360/∞ will span the circle makes the implicit assumption that every real number has another unique real that is "next" to it, so that you could span all possible rotational angles by stepping from one real to the next until you've stepped through them all.

Unfortunately, compelling as this intuition may be, it does not work mathematically. The very definition of the real numbers means that between every two reals there must lie another real (and in fact, continuumly many reals!). So no matter how small our step is, we will always miss some reals. (Actually, that is the understatement of the century: any non-zero step skips over continuumly many reals---this is way, wayyy, wayyyyyyy more than the number of naturals, which is infinite!) Of course, here we're talking about infinitesimals, but this still can't be done. Suppose we expand the reals by allowing an infinitesimal element L. Then we could start with a real number r, and step from it to r+L. However, r+L is not a real number (the only time an "expanded" number r+kL is real is when k is zero, since we can fit infinitely many infinitesimals between two reals). What about r+2L, r+3L, ... etc.? None of them are real numbers. This means that we have not made very much progressive from r to the "next" real number. We've barely begun to approach the "next" real! But what if we add infinitely many L's? Well, that depends on what size L is. It also depends on what kind of infinity we're talking about (remember, there are multiple incompatible infinities). To make the following discussion easier, let's call "numbers" of the form r+kL "expanded numbers", and we say r is the "real part" (the part corresponding with a real number), and kL the "infinitesimal part" (corresponding with the infinitesimal distance from the real number r to the expanded number r+kL). Our task boils down to essentially finding a k such that the real part of (r+kL) is greater than r (since otherwise, we have made no progress at all). One possible infinity, given that we've expanded our reals with L, is simply 1/L. But 1/L is "too big": (1/L)*L = 1, which has already crossed continuumly many reals (it stepped from r to r+1, which is not what we want, since it skipped over all the reals between r and r+1). Is it possible to use a smaller infinity, say K, so that by taking K steps of L we will get to the "next" real number? Unfortunately, the answer is no, because we come right back to the original quandry: the only way for an infinitesimal to make any real progress from r is if L is multiplied by something that will give it a non-zero real part. But what should this non-zero real part be? No matter how small we make it, say KL = s for some small real number s, there is a real number between r and r+s, so our step is still too big: it has missed this real that lies in between. This is true for all possible real numbers s, so we're stuck: if we chose K such that KL has a zero real part (i.e., KL is still an infinitesimal), then r+KL is smaller than all real numbers greater than r, so we've made no progress at all. But if we chose K such that KL does make some progress, it always turns out to be the case that it's too much progress: r+KL is too big, and has skipped over continuumly many reals that lie between r and r+KL.

These are the only possibilities, so it is actually impossible to span the reals by applying an infinitesimal step each time! In other words, there is no such rotation angle as 360/∞, even if we allow infinitesimals!

The bottomline is that this is just a manner of speech that hides the fact that we're taking a limit here (the limit of all polygons rotated n times by 360/n, as n diverges without bound). The limit, again, stands apart from the approximating sequence.

This can be seen most clearly in the considerations on your webpage, which notes that in the sequence of polygons with increasing degree, if we keep the radius fixed, it will converge to a circle, and if we keep the edge length fixed, it converges to an apeirogon. There is no reason to think that these two limits are the same thing, because, after all, they are the limits of two different sequences!
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Re: points in a line

Postby Eric B » Thu Nov 06, 2008 2:16 pm

quickfur wrote:
Eric B wrote:
quickfur wrote:
In fact, this has been studied before.
Oh, by who, and where can I find info on it. Is there a name for this branch of math?

It's not really a branch of math per se, it was just some random musings by a mathematician.
You earlier mentioned Riemann. Is that who it is? I'll have to look him up. From him, I'm familiar with his models of the shape of the universe.
The properties of such "numbers" do not lend themselves to very many useful applications (even within a pure math context), so they aren't really worth studying very much.
Yeah; like I said; the only thing you could do with them is add and subtract. It would only only serve an alternate negative number system for subtraction. I imagine that that's not really needed, but it's still an interesting extension.

Only, the set of such "numbers" have rather strange properties (unfortunately I can't recall them off the top of my head) that makes them difficult to map back to the usual numbers.
This is probably like what I mentioned. Like ...99999. not being divisible by 9, even though it consists of all 9's. And that multiplication and division become meaningless in that range (Because where 0 is the center of symmetry for addition; 1 is the center of symmetry for multiplication).

The usual term is "identity".
"What's "identity"? The number at the center of symmetry? (I had forgotten what the term was).

As for divisibility, ...99999 is definitely divisible by 9: it divides evenly into ...11111. It should not be surprising that an "infinite" number divides into another "infinite" number. :-)
But what I meant was that it's not evenly divisible, like 99999/9. After all, it is -1/9, which is -.111111..., which isn't considered "going into" the number.
That's interesting that both are an infinite sequenceof 1's, though on opposite sides of the decimal point (mirror image). So the one on the left side looks just as much like a whole number as the new form of -1, even though it isnt. I guess that's another of those unusual properties.

Likewise, with the star analogy, as n approaches ∞, for any the so-called "points" flatten out towards the surface; hence, all finite densities are collapsed onto the perimeter. So yes, the circle would represent all finite Schlafli densities (I also looked into what the "highest" density would be. Of course, it would also be infinite, but it would swell either into an infinitely dense solid disk (If the total diameter is fixed), or swell outward to infinity; still with infinite density (if the radius is fixed). An infinite sequence of polygons rotated at 360/∞ would produce produce an infinitely dense ring, and the symbol would still be {n/∞}.

Of course, all of this is abstract, and not "real" in our everyday sense. So I am quite aware of the limitations.

Not sure what you mean by "rotated at 360/∞"---presumably an infinitesimal rotation that eventually spans all possible rotations about the origin. Again, this is where one has to be careful about generalizing finite concepts into infinity. Saying that an infinite number of rotations about 360/∞ will span the circle makes the implicit assumption that every real number has another unique real that is "next" to it, so that you could span all possible rotational angles by stepping from one real to the next until you've stepped through them all.

Unfortunately, compelling as this intuition may be, it does not work mathematically. The very definition of the real numbers means that between every two reals there must lie another real (and in fact, continuumly many reals!). So no matter how small our step is, we will always miss some reals. (Actually, that is the understatement of the century: any non-zero step skips over continuumly many reals---this is way, wayyy, wayyyyyyy more than the number of naturals, which is infinite!) Of course, here we're talking about infinitesimals, but this still can't be done. Suppose we expand the reals by allowing an infinitesimal element L. Then we could start with a real number r, and step from it to r+L. However, r+L is not a real number (the only time an "expanded" number r+kL is real is when k is zero, since we can fit infinitely many infinitesimals between two reals). What about r+2L, r+3L, ... etc.? None of them are real numbers. This means that we have not made very much progressive from r to the "next" real number. We've barely begun to approach the "next" real! But what if we add infinitely many L's? Well, that depends on what size L is. It also depends on what kind of infinity we're talking about (remember, there are multiple incompatible infinities). To make the following discussion easier, let's call "numbers" of the form r+kL "expanded numbers", and we say r is the "real part" (the part corresponding with a real number), and kL the "infinitesimal part" (corresponding with the infinitesimal distance from the real number r to the expanded number r+kL). Our task boils down to essentially finding a k such that the real part of (r+kL) is greater than r (since otherwise, we have made no progress at all). One possible infinity, given that we've expanded our reals with L, is simply 1/L. But 1/L is "too big": (1/L)*L = 1, which has already crossed continuumly many reals (it stepped from r to r+1, which is not what we want, since it skipped over all the reals between r and r+1). Is it possible to use a smaller infinity, say K, so that by taking K steps of L we will get to the "next" real number? Unfortunately, the answer is no, because we come right back to the original quandry: the only way for an infinitesimal to make any real progress from r is if L is multiplied by something that will give it a non-zero real part. But what should this non-zero real part be? No matter how small we make it, say KL = s for some small real number s, there is a real number between r and r+s, so our step is still too big: it has missed this real that lies in between. This is true for all possible real numbers s, so we're stuck: if we chose K such that KL has a zero real part (i.e., KL is still an infinitesimal), then r+KL is smaller than all real numbers greater than r, so we've made no progress at all. But if we chose K such that KL does make some progress, it always turns out to be the case that it's too much progress: r+KL is too big, and has skipped over continuumly many reals that lie between r and r+KL.

These are the only possibilities, so it is actually impossible to span the reals by applying an infinitesimal step each time! In other words, there is no such rotation angle as 360/∞, even if we allow infinitesimals!

The bottomline is that this is just a manner of speech that hides the fact that we're taking a limit here (the limit of all polygons rotated n times by 360/n, as n diverges without bound). The limit, again, stands apart from the approximating sequence.
This can be seen most clearly in the considerations on your webpage,
Yeah; that is basically what I was conveying by saying that there was "no next to" a given point, and such.
However, if you were to take a polygon and rotate it 360° (or even 360/n), it would sweep out the disk I mentioned; covering all the infinitessimal points or real numbers inbetween.
which notes that in the sequence of polygons with increasing degree, if we keep the radius fixed, it will converge to a circle, and if we keep the edge length fixed, it converges to an apeirogon. There is no reason to think that these two limits are the same thing, because, after all, they are the limits of two different sequences!

When I say "same thing", I mean, hypothetically, an ∞-gon. (Maybe I shouldn't say "same thing") Yes, it's two kinds of infinity; but still both are the limit of taking a polygon and extending n to infinity. So there is some sort of hypothetical connection between them. They're both different perspectives (I guess a better word than "magnification") of the same result of the same action.
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Re: points in a line

Postby zero » Wed Apr 08, 2009 6:57 am

wendy wrote:A line, then is not "made of" points, but "contains" points. There are places on the line that have no construction.

Nicely put.
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