Also, how do you get out of Hilbert's Hotel? In Computer Science there are two types of data structures called stacks and queues. With a queue you can take your data back out only from the oposite end from which it entered, a first-in-first-out (FIFO) structure. With a stack you put data in and the only way to get it out again is to start with the last one in. This is the Holy data structure where the first shall be last and the last shall be first (FILO). I imagine getting out of the Magical Realm of Infinity is like that. Perhaps I should call it the Heavenly Realm of Infinity.

]]>Will, one thing worth noting is that you sometimes have to be careful with wiki syntax. For example, if you type `--stuff--`, you get stuff (a strikeout). So, when you typed your string of dashes in your first post, you inadvertently ended up striking out quite a few lines. If you take the time to learn some $\LaTeX$, you'll see how powerful it can be in these situations, where ordinary text just ain't cutting it. Again, you can learn more about $\LaTeX$ on the quick LaTeX guide page.

Euclid said, instead of an infinite number of primes there are more primes then in any collection of primes. The Indian Mathematical text Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

Enumerable: lowest, intermediate, and highest

Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable

Infinite: nearly infinite, truly infinite, infinitely infinite

In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asa khyata ("countless, innumerable") and ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.

In a paper called Leibniz and Cantor on the Actual Infinite by Richard Arthur (http://www.humanities.mcmaster.ca/~rarthur/papers/LeibCant.pdf)

Arthur discusses Cantor’s view that there exists a number r that is one less then the infinite number z (The article by Richard Arthur used small omega) where z is an infinite quantity and r is the number of times z can be divided. He talked about r being a finite number because the component parts are finitely divisible into an infinite number of parts “Cantor’s definition z is the first ordinal number > r for all finite numbers r.”

Cantor also used other terms to describe values as uncountable or innumerable, etc which seem to me to make a distinction from these type of numbers and actual infinite numbers. At least that is the impression I got from my reading of “Godel’s Proof” by Ernest Nagel and James R. Newman.

So how to calculate it? We can think of .999… as the infinite series 9/10 + 9/100 + 9/100 + … + 9/10^n for n -> infinity. We can show that 1 - this sum converges to 0. So the difference between 1^2 and .999…^2 is 0, and we can take .999…^2 to be 1 (it differs from 1 by 0, so why not?).

Your argument is actually very close to Aristotle's argument that there cannot be any real infinities, only "potential" infinities that go on "as long as you like." Zeno, who agreed with Aristotle on this, in fact showed the opposite: that infinities are as "real" as anything else we reason with, but behave differently in some situations than "finities" like 1 or pi. Achilles does catch the tortoise, despite having to pass through an infinite number of half-intervals to do so, and we can calculate precisely how that series of steps converges, and so when he will take the lead.

Infinities are not magic, just un (in) limited (finite) in some direction. It is not a Roach Hotel, it is Hilbert's Hotel: when a new guest arrives, there is always room despite the "No Vacancy" sign: the clerk just calls Room 1 and tells the occupant to move up one room, and tell the occupant of room 2 to do likewise.

]]>\begin{align} \begin{list_type*} \item \item 9 \item X 0.9999999 \item -------------- \item 8.1 \item 0.81 \item 0.081 \item 0.0081 \item 0.00081 \item 0.000081 \item + . . . \item --------------- \item 8.999991 \end{list_type*} \end{align}

One can see that carrying this operation out forever will lead to an infinite string of 9s. We’ll just forget about the 1

because numbers disappear in Infinity. Infinity is a magical realm. Numbers go in but they never come out. Like

the Roach Hotel of Mathematics.

How about if we square .9 and square 1.

We can not do the operation out because we can’t see what happens at the right end of the string where it exists

in Infinity. So, let’s try it in the finite realm and see if we can tell what is happening in Infinity.

.9 x .9 = .81 No 9s in the resulting string of numbers and .81 is obviously not equal to .9.

How about:

$.99^{2} = .81 + .081 + .0891 = .8 + .17 + .010 +.0001 = .9801$ One 9 out of 4 digits but it’s a leading 9.

(I wanted to do this out to show all values are present and accounted for).

Ok, next problem: (Assume the same process of calculation for all that follows)

.999^2 = .998001 More leading 9s. However there are still less 9s then other numbers.

Again:

.9999^2 = .99980001 Wow, I see a pattern here. Now let’s assume this process goes on forever.

There will be just as many 0s as 9s and the 9s will be 1 less then the original

number of nines. Also, there are always 2 more digits that are neither 9s nor

0s then making the string of digits of non-9s two longer then there are 9s.

Now here’s the thing about magic realms. All travelers arrive at once. None who leaves for a magic realm can get there before another as long as they left at the same time. There is no closer or farther away. Not only will the length of the string of numbers reach infinity but the 9s will get there at the same time and so will the 0s. You can’t get half way to infinity. You are either there or not. So, there are not ½ - 2 to Infinity of 9s. And, as I already stated about the magic realms including Infinity, once you go in, like magic, you disappear. Only the 9s can be found because they are behind in the journey to infinity. Although, they arrived there at the same time as the 1s, they are still last and push all the numbers ahead of them through the gates where they are lost forever, like magic. The nines have a foot hold in the finite realm and unless another traveler on its way to Infinity, and I might point out that they have to also be on the same path, starts later then the 9s, they will remain in the Finite realm. There is no indication that anyone else wants to travel this path to Infinity so the 9s are all that are left. Does that make sense? Magic!

Wait, wait, How do I know the 9s as well as the other numbers actually make it to Infinity? It’s a long long long way. They can’t make it on their own. Don’t I or someone have to keep doing to calculations never endingly. I got it! Let’s not and say we did. Even still. Any teacher would know I was lying. “You never did the calculations out!”

I would say, “Well… Ah, I did do the calculations, teach.”

“No, I don’t thinks so. I don’t know how you got those 9s to Infinity but I know you didn’t do all the calculations. Now did you?”

“Well… No, not all of them. But I did most of ‘um. Really, I did nearly all of ‘um.”

“I am unconvinced. You couldn’t have done that many in the time you had.”

“Ah… um, at least half of them. I did them ‘till the 8 got to infinity, then I had to go to bed. That‘s 1 more then half.”

“Son, if you did that many calculations then you would know that the 8 as well as the 1 never get to Infinity.”

I defend myself, “Yeah they do. They are gone, aren’t they? They disappeared once they got to Infinity.”

The teacher is really angry now, “No no no no, you didn’t do all the calculations and they can’t just disappear. What did you do with the 8 and the 1. There’s only one of each so you must have stashed them somewhere. I think you need to go into the corner and think about where those missing numbers are.”

What could I do but sit in the corner. I didn’t know where they went. I still don’t.