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.

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.

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?

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.

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.

plot(f(x),(-5,5))

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).

$\int_0^1 \sin x dx$

f(x)=x^2

integral(f(x),x)

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.

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

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.

Here is an interesting thought about the Natural numbers; there is no natural number that is of infinite value. All Natural numbers are finite, even though the set of all Natural numbers is infinite. Correct me if I'm wrong, but the cardinality of $\mathbb{N}$ is greater than any element of $\mathbb{N}$ because the cardinality of $\mathbb{N}$ is $\aleph_0$ and $\aleph_0$ is an infinite cardinal number.

Picky semantics: $\aleph_0$ is a cardinal for an infinite set. Do you see the distinction?

This seems an important distinction, when talking about numbers that go to Infinity. If, for example, $n\in\mathbb{N}$ and $n\to\infty$, then we know that $n$ never actually gets to infinity. That is, $n$ is never infinite in value, but there is always another $n+1$ to become the next $n$.

Maybe put quotes around the last $n$ since you are abusing notation: but there is always another $n+1$ to become the next "$n$".

The significance of this, for me here, is when applying the idea of breaking an infinite set up into a set of infinite sets by the property I talked about above, then how do we move between subsets of $\mathbb{N}$ if they are infinite but the contiguous neighboring set is filled with all finite elements.

I don't know what "contiguous neighboring set" means here. Does contiguous mean consecutive here?

We can't. We can't divide an infinite set of finite values into contiguous infinite sets. Otherwise, the first element of the second set would already be outside the Natural numbers. Therefore, to divide $\mathbb{N}$ into an infinite set of infinite sets the first element, as every element, would have to be a finite element and the only way that could be the case is if we did something like taking every element of $\mathbb{N}$ in turn and assigning it as the first element of the next set. But then, how would we ever get the second element of $\mathbb{N}$? Perhapse $\mathbb{N}$ can't be divided into an infinite set of infinite sets.

I'm not following this.

I don't see any problem with taking every second element of $\mathbb{N}$ and putting it into a separate set and saying that the two sets are of equal and infinite size. That does seem to hold with my understanding of the nature of $\mathbb{N}$ as an infinite set of finite elements.

Sure, this is fine.

It seems ok to then take every other element of the resulting two sets to make four sets of equal size and they should also be of cardinality equal to $\aleph_0$.

You need to be careful with the phrase "equal size" here.

It seems that this process can go on indefinitely but what happens to the size of the elements in each set as you begin to count through them?

If I'm following you (and I might not be), at each step you are chopping an infinite subset into 2 nonempty disjoint infinite subsets. So, what do you mean "begin to count through them"?

When the cardinality of the supper set of infinite sets reaches $\aleph_0$ then how could there be more then one element in each set?

You meant "super set" instead of "supper" (like dinner). However, that's not the correct phrase here. A super set is the set of all subsets of the set in question and that's not necessarily relevant to this discussion. You're happening upon some of the interesting behaviors of infinite sets. Here's another. Each point on the real line has zero length. However, the sum total of all the points in the interval $[0,1]$ has length 1. Wrap your head around that.

This seems an important distinction, when talking about numbers that go to Infinity. If, for example, $n\in\mathbb{N}$ and $n\to\infty$, then we know that $n$ never actually gets to infinity. That is, $n$ is never infinite in value, but there is always another $n+1$ to become the next $n$. The significance of this, for me here, is when applying the idea of breaking an infinite set up into a set of infinite sets by the property I talked about above, then how do we move between subsets of $\mathbb{N}$ if they are infinite but the contiguous neighboring set is filled with all finite elements. We can't. We can't divide an infinite set of finite values into contiguous infinite sets. Otherwise, the first element of the second set would already be outside the Natural numbers. Therefore, to divide $\mathbb{N}$into an infinite set of infinite sets the first element, as every element, would have to be a finite element and the only way that could be the case is if we did something like taking every element of $\mathbb{N}$ in turn and assigning it as the first element of the next set. But then, how would we ever get the second element of $\mathbb{N}$? Perhapse $\mathbb{N}$ can't be divided into an infinite set of infinite sets.

I don't see any problem with taking every second element of $\mathbb{N}$and putting it into a separate set and saying that the two sets are of equal and infinite size. That does seem to hold with my understanding of the nature of $\mathbb{N}$as an infinite set of finite elements. It seems ok to then take every other element of the resulting two sets to make four sets of equal size and they should also be of cardinality equal to $\aleph_0$. It seems that this process can go on indefinitely but what happens to the size of the elements in each set as you begin to count through them? When the cardinality of the supper set of infinite sets reaches $\aleph_0$ then how could there be more then one element in each set?

Thanks - Ashley

We can thus consider the set of all sets and subsets of an infinite set as having a cardinality equal to the cardinality of the base set (in this case, the Natural numbers set N). But, what about that 1-1 correspondence between the set of all Real numbers to the set of all Natural numbers? If we can divide the set of all Natural numbers up into an infinite number of subsets whos cadinality is equal to the parent set then how can we fail to find a corresponding element in the set of Natural numbers to the set of Real numbers? I have shown that I can divide the Natural numbers into an infinite set of infinite sets. Further more, each subset of infinite cardinality can contain completely unique elements from any other division of the parent set. So, before I have even made a dent in exhausting the elements of the Natural numbers, I can divide off a subset that could itself be divided off into a collection of infinite sets just to cover the elements between 0 and 1 in the Real number set. If there aren't enough, I can divide that sub sub sub set again and still have an infinily divisible set to cover the correspondance with. I would never lack for a unique element from the Natural numbers set to match up with an element to the Real numbers set.

I know that the numbers in the Natural number set would become absolutly enormous by the time we covered the interval from 0 to 1 in the Reals but, so what, I still have plenty to move to the next interval. In fact I still have the same number of elements in the Natural numbers that I had before I used up an infinitely small fraction of them to cover all previous intervals. I am not going to run out of Natural numbers by placing them in a 1 to 1 correspondence with any other set!

Here are some potential questions to address:

1. What is a fractal?
2. What are some properties that fractals possess?
3. What is the Mandelbrot set?
4. What points are in versus not in the Mandelbrot set?
5. Who is Benoit Mandelbrot?
6. What makes fractals interesting?
7. What are some applications for fractals?
8. What parts of the movie were interesting or intriguing?
9. What parts were confusing?
10. Did the movie make you think of questions that the movie did not address? If so, what were they?

Let me know if that isn't enough to get you started.

That is rather cool. I hope it's important.

I hope that made sense; are we headed in the right direction.

If $v(t)$ is a velocity function, then the distance travelled during $t=a$ to $t=b$ is the same as the area under the curve of $v$ over $[a,b]$.

Example 7.1 in Section 7.1 talks explicitly about this and I will say a little more about why this the case at the beginning of class tomorrow (Monday). To attack these problems, you really need to find the area under the curve between the specified $t$ values. For example, in Exercise 7.1.1, if you find the area under the curve over the interval $[1,2]$, then you will know exactly how far the object travelled during this interval. Then you can just add this number to 5 (the given position at $t=1$) to determine where the object is at $t=2$. Finding the area under the curve is easy here (no rectangle approximations required) since the graph of the velocity function is just a straight line (think triangles and rectangles).

I hope that helps.