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So, is there a difference between the number of 9s in 0.9~ and the number of elements in $\mathbb{N}$? While it is accepted that there are an infinite number of elements in $\mathbb{N}$, there really are just more elements in $\mathbb{N}$ then in any collection of elements from $\mathbb{N}$ because every collection of the elements of $\mathbb{N}$ will always be finite. $\mathbb{N}$ is open and unbounded to the right but it is always finite in that no element of $\mathbb{N}$ is infinite in value. However, the number of 9s in 0.9~ needs to be infinite, not open and unbounded, for it to equal 1. I don't believe these two ideas are the same. An unbounded set of finite elements can not be infinite but it is always growing as there is always more to include. They are never ALL there, but the 9s in 0.9~ have to be all there for it to equal 1.

I don't feel very clear on this but I do see the glimmer of an idea that seems to have merrit. There are differences to infinite vs unbounded. One feels complete but incomprehensible while the other is never realized but very comprehensible.

For an example of the type of model I am picturing; I mentioned earlier a Poincarè model of hyperbolic space where the space is represented by a bounding circle but the circle itself is understood to be outside the unbounded space. All lines in B-L (Bolyai-Lobachevsky) space
represented by the Poincarè model are considered infinite in length. They "intersect" the bounding circle at a ninty degree angle and at that point they may converge at an angle of zero degrees to one another. Of course, any segment of their line that is not at the bounding circle is considered of finite length and the angle of converging lines becomes greater then zero. There would then be a correlation between the angle of intersecting lines with the finite length of the segment. That is, the closer to the center of the bounding circle a line segment is, the closer to a direct relationship the model is to that segments length. The closer to the boundry, the lengths become hyperbolic in size (an inch is far far greater, until at the boundry, it is infinite). For a similar line that represents the numberline from some finite point to Infinity, the image is not quite like that, although that might make more sence. For any finite value on the line, there is no direct representation of its position, in relation to another finite value other then the one point. Such a line can only, really, show two points; those that are finite and those that are Infinite. Any halfway or 1/4 way or 1/8th way, etc., would be equal to every other point that was infinite and therefore, not an element of $\mathbb{N}$. I can tell you anything I like about the numbers but I can't show it to you in such a model. I could define a model that progressed towards the boundry of Infinity in such a way that it would never get there; 1" beween the first two numbers, 1/2" between the second and third numbers, 1/4" between the third and forth numbers, and so on, but it would be impossible to show the position of very many numbers before they all looked like they existed at Infinity. Obviously the number line would only be two inches long and that is not very long for a model. This is just a visuallization excersize that makes me realize that when talking about finite versus infinite, there is little difference between that and a binary system.

If you consider any finite collection of points in relation to all the Natural numbers, no matter how many are in the collection, the proportion of the number of elements in the collection to the whole

It doesn't make sense to talk about a proportion involving an infinite set. However, you are noticing the difference between being finite and infinite.

the collection of points would appear to occupy no more of the number line then any other collection of points no matter how few were in that collection.

You need to be careful here. If you have a collection of 5 consecutive natural numbers, then the amount of 1-dimensional space these points take up is 0. However, they occur in an interval that is 4 units long. Are you thinking of the natural numbers embedded in the real line or just a set devoid of geometry?

Thus you would be unable to distinguish one point from a collection of a thrillion points when considering the entire number line. Any finite collection would appear as dimensionless as a single point.

Based on the fact that you can tell me that there is 1 point versus a trillion points, I think you can tell them apart. However, as sets embedded in the real line, they have measure 0.

If you consider any finite collection of points in relation to all the Natural numbers, no matter how many are in the collection, the proportion of the number of elements in the collection to the whole, the collection of points would appear to occupy no more of the number line then any other collection of points no matter how few were in that collection. Thus you would be unable to distinguish one point from a collection of a thrillion points when considering the entire number line. Any finite collection would appear as dimensionless as a single point.

since the collection of points makes a number line that is infinite in length, then looking at the whole line one can not see individual points.

I don't understand this. There are some subtle issues to sort out here. What exactly does "looking at the whole line" really mean? I can consider the whole line, but I can only see a finite chunk of it at a time, and when I do this, I can see individual points just fine (okay, I can't see the points, but I can see the "fat" labels we put on them).

Therefore all finite points, in relationship to the line as a whole would appear to occupy the same space and appears to be continuous.

Huh?

I don't think I'm contradicting myself. I'm just saying that, since the collection of points makes a number line that is infinite in length, then looking at the whole line one can not see individual points. To do so would give a visual position to that point that was unique and allow you to make progress towards an end that does not actually exist. Therefore all finite points, in relationship to the line as a whole would appear to occupy the same space and appears to be continuous. That doesn't change the fact that we know it's not a continuum.

We know, too, that it isn't really a line but a series of discrete numbers or points

Yes.

but when looking at the whole collection of points it looks like a continuous line. No matter how good our eye sight is, we can not distinguish individual points.

No. This contradicts the first claim.

Image, if you can, the Natural number line in its entirety. On the left endpoint is One. To the right, from there, is a line of infinite length. We know, too, that it isn't really a line but a series of discrete numbers or points, but when looking at the whole collection of points it looks like a continuous line. No matter how good our eye sight is, we can not distinguish individual points. Perhapse it would help to picture the line inside a circle where the circle, all points at $n$distance from the center represents the unlimited boundry of numbers, like the boundries of a Poincarè model of Hyperbolic space. The center of the circle would then be Zero, the Origin. The Natural number line, taken as a whole, would go from the center to a point near the bounding circle such that, no matter how close one zoomed in it would look like it touched the boundry. Whether or not it actually does is unimportant, for now. However, that it does not touch the actual center is somewhat important. We know, by definition, that the first point of the Natural number line is one whole number away from the Origin, but when looking at the number line as a whole, that fact is obscured and we can draw the line as if it were eminating right from the center. We can do this because the proportionate difference between Zero and One to the length of the entire number line of all Natural numbers is infinitesimal.

Now that we have a visual model for the Natural number line; what can we do with it to help describe an infinite set of finite numbers? For one thing, we can see that the difference between One and Zero on this number line does not help in locating the finite beginning. We must use what we know about the Natural numbers to find that, but what about One-Million? can we see the difference there? Nope! If you consider that there are still an infinite number of points to the right of One-Million, then we can't visually tell the difference between One-Million and the Origin or Zero. The same can be said for One-Googol or One-Googolplex or even Skewe's Number $10^{10^{10^{1000}}}$.

There is still an infinite number of points to the right of these numbers and these numbers are all finite so they still sit, visually, at the Origin. In fact, that is something I find entirely fascinating. No matter what number we pick, it will be finite and there will always be an infinite number of points to the right of it. Therefore, in our visual model no points on our line can exist anywhere along the line except at Zero, yet the line to Infinity must also exist in order for there to be an infinite number of points that never leave the Origin. We can draw a point at the midpoint of the line that is $n/2$ from the center of our circle and that point would exist on the Natural number line but it has to be an infinite number and is therefore not a Natural number. No mater what, if all Natural numbers are finite, then all Natural numbers go no where along the Natural number line. They don't get any closer to Infinity, they don't get infinite and they can't become distiguishable from Zero in relationship to the number line as a whole.

Re: final exam - Sage by Ian RandIan Rand, 17 Dec 2010 00:32

That's exactly what I have, I've defined the function, I then factored it, then showed it but when I go to plot it I only get Traceback stuff. Hmm…..

Re: final exam - Sage by Ian RandIan Rand, 17 Dec 2010 00:21
Re: final exam - Sage by Dana ErnstDana Ernst, 17 Dec 2010 00:18

haha just got it before you put it up.

One last question I've defined f(x) for the first couple of questions but I'm having a hard time plotting the functions from (-5,5).

Re: final exam - Sage by Ian RandIan Rand, 17 Dec 2010 00:15

To get $\int_0^1 \sin x dx$, you would type:

$\int_0^1 \sin x dx$

Re: final exam - Sage by Dana ErnstDana Ernst, 17 Dec 2010 00:11

First, you need to define the function. For example:

f(x)=x^2

integral(f(x),x)

Re: final exam - Sage by Dana ErnstDana Ernst, 17 Dec 2010 00:10

sorry i meant the function sin(x) from the interval (0,1) how would that look in latex?

Re: final exam - Sage by Ian RandIan Rand, 17 Dec 2010 00:09

What if I want to integrate the function (f(x))?

Re: final exam - Sage by Ian RandIan Rand, 17 Dec 2010 00:06

Here is an example of how to do an indefinite integral using Sage:

integral(x^2,x)

The first term is the function that you are integrating (in this case $y=x^2$). The second term is the variable that you are integrating with respect to (in this case $x$; this represents $dx$). Here is an example of a definite integral:

integral(x^2,x,0,1)

The 0 and 1 are the lower and upper limits, respectively. If you want to type this in $\LaTeX$, you would type:

$\int_0^1 x^2 dx$

This looks like: $\int_0^1 x^2 dx$. I hope that helps. If you have more questions, please let me know.

Re: final exam - Sage by Dana ErnstDana Ernst, 16 Dec 2010 14:07

Leigh Anne,

You have a typo on your 4th term. The coefficient should be 114 instead of 11. That should fix your problem.

Re: final exam by Dana ErnstDana Ernst, 16 Dec 2010 13:56

I'm trying to type in a problem on sage but I cant remember what the write abbreviation or anything is or how to look at them all. I want to intergrate a problem. Does anyone know the right wording for that?

final exam - Sage by Tabby MarcouxTabby Marcoux, 15 Dec 2010 20:25

We can't divide an infinite set of finite values into contiguous infinite sets.

By this I mean that the value of each element in the previous set is strictly less than any value of the elements in the next set. In finite sets the collection of sets would look like: {{1,2,3}, {4,5,6}, {7,8,9}, … . But, of course, for any set of infinite sets that are contiguous like that, there is no last value in the previous set such that its following set can start with the next value and continue on. Therefore, $\mathbb{N}$can't be divided into a collection of infinite subsets in this way. That's not to say that $\mathbb{N}$ can't be divided into infinte disjointed sub-sets.

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