Zeno's Paradoxes

DICHOTOMY: It is impossible to cover any distance, because half the distance must be traversed first, then half the remaining distance, then again half of what remains, and so on. Some portion of the distance to be covered is ALWAYS left to cover. Therefore, motion is impossible.

ACHILLES AND THE TORTOISE: Achilles runs to overtake the tortoise, but he must reach the point where the tortoise started, from which it has already departed. Repeating indefinitely, Achilles gets to each new point in the race, the tortoise having been there, has already left. Therefore, even though Achilles is much faster than the tortoise, he can NEVER even tie.

THE ARROW: An arrow shot from a bow must be moving at every instant in its flight. But at every instant it must be somewhere in space. However, if it is always in some one place, it can't be in transit at every instant, for to be in transit is to be NOWHERE

http://scidiv.bcc.ctc.edu/Math/Zeno.html

The Story Of The Hotel Ad Infinitum

This story is not true (in the sense of being real ), for certainly there is no such thing as an infinite hotel . What I have done is taken an idea of David Hilbert's [1862-1943] and put it in a context that students would enjoy, so rest assured that the mathematics is perfectly valid. The point of the story is that the concept of infinity is a very strange and abstract thing. So if you'll just play along with me, I'll tell you about a very weird night I had one time while I was in college, working at the Hotel Ad Infinitum ...

I arrived at work that night, ready to relieve the desk clerk who worked before me on Friday nights. He told me the most unbelievable thing: the hotel was full! Perhaps I should describe the place to you. It was just one great big long hallway; there was a door at the entrance, and when you walked in, the desk was at the left. Then the hall opened before you, endlessly. Along the left hand side of the hall were all the odd numbered rooms {1, 3, 5, 7, 9, ...} and at the right side were the even numbered rooms {2, 4, 6, 8, ...}. The hallway went on and on, on and on forever! It was hard to imagine that the place was full, but he assured me that it was. I should have known right then and there that something strange was going on, but I had an exam coming up, so I sat down, pulled out my calculus book, and started studying.

A little after one o'clock, a huge stretch limo pulled into the parking lot. A chauffeur got out, and walked in.

"Howdy, I need a room for the night; my boss is sleepy; he had a hard game tonight."

"Baseball player?" That figured; even in those days salaries were out-of-sight! But I told him that the place was full: "That's what the sign says, right?"

"Wrong; back in a minute." He went out to the limo, popped open the trunk, and pulled out a little package about the size of a loaf of banana nut bread; it turned out it was a different kind of bread all together! He brought it in, set it on the desk and slid off its velvet cover and--lo and behold--it was a gold brick!

We'd been studying compound interest in one of my classes, and I knew that the student loans I was taking out were going to cost me a LOT more than I was getting from them. My eyes widened with amazement. I looked up at the driver, who was smiling as he said, "So, you think there's something we can work out?"

You bet there was! I immediately grabbed the intercom, and announced to all the guests: "Please excuse the interruption, but if you're in Room N , would you kindly move to Room N+1 ?"

So, the guy in Room1 went to Room 2, the couple in Room 2 went to Room 3, et cetera. It was a mad flurry of rushing folks, dashing across the infinitely long hallway at the Hotel Ad Infinitum ... amazing! Please note that no one lost out, because there was no end to the hallway, and when everyone was settled, there was no one in Room1, right? So the baseball guy took Room 1, I took the gold brick, and proceeded to write my letter of resignation. Incidentally, this must mean that infinity plus one equals infinity, because I took an infinite number of quests, added the baseball player, and put them all up in the Hotel Ad Infinitum. Amazing, isn't it? I was blown away, but the real weirdness had not yet begun.

While I was trying to figure out how to turn my gold brick into normal money, I heard a tremendous rattling sound, looked out into the parking lot, and suddenly there appeared a beat up old VW van, smoke pouring out of its engine, a little trail of oil following it. The driver turned it off (though it kept running for a bit, sputtering and clicking and gasping) and ran into the hotel. Looking a little wild-eyed, he exclaimed that they needed rooms for the night. They? Rooms?

"Sir, did you notice the NO VACANCY sign outside, all lit up, bright, flashing neon?"

He talked on for quite a bit, got confused a few times, but I managed to sort out the story. Seems Dylan was playing nearby, and the van outside was carrying an infinite number of Dylan freaks, all ready to catch their man in action. Actually, he was my man too, so I was very interested. We talked awhile and it turned out that he had an extra ticket. I was wondering where he had gotten an infinite number plus one Dylan tickets, but I figured what-the-hey? Anyway, he offered to lay the ticket on me if only I could put 'em up for the night. I was ready to quit anyway, so I figured why not, and jumped back on the intercom and announced, "Ah, sorry to interrupt again, folks, but we have an emergency here, and if you're in Room N would you please move to Room 2N."

So the baseball player went to Room 2, the guy in Room 2 went to Room 4, the couple in Room 3 went to Room 6, and so on. Again, no one was put out on the street, since--as you may have guessed--the Hotel Ad Infinitum had no back door! When that was over, all of the original guests, along with the baseball guy, were all on the right-hand side of the hotel, in the even-numbered rooms (of which there are an infinite number) and that left all the odd-numbered rooms vacant. So, I put the Dylan freaks into the odd-numbered rooms, which was sort of appropriate, I suspect! I guess this means that infinity plus infinity equals infinity, since I added an infinite number of Dylan freaks and put them in an infinite hotel which was already full!?!? Wait a minute ...

So there I was, my gold brick, resignation letter, and Dylan ticket in hand, staring at the clock, counting down to my new-found freedom, when all of a sudden--oh no, how could this be--a caravan of buses pulled in, an infinite number of buses, and on each bus, an infinite number of people! An infinite number of infinities! What was happening, as I was soon to find out, was that there was to be an ecumenical council of all the galaxy's religions, and every single religion had sent its own busload, loaded with an infinite number of its faithful! Yes, I was seriously in trouble on this one! Naturally, the driver of the first bus jumped out, came bounding in, and requested "a few" rooms for the night ... uh huh! Sure, an infinite number of infinities, it was clear to me, clear as mud! Of course, I reminded him about the sign and how we were full and all, and he smiled and began asking about the Dylan freaks, about the baseball player (how he knew I had no idea) and then started to remind me of the story of Mary and Joseph trying to get a room at the inn, and suddenly it occurred to me that with an infinite number of religions being represented here (all the religions of the galaxy) that one of them, no doubt, was the "right" one, and that it would not be wise to go down as the guy who wouldn't give them a room for the night and sent them to the manger. I mean, did you ever wonder about that guy that sent Mary and Joseph to the manger? I wonder how he's doing?

Well, in one of my courses, we'd been studying prime numbers {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...} and how Euclid, nearly two-thousand years ago, proved that the list of prime numbers is infinitely long. So I got the following idea: I got back on the intercom (last time, I promise) and asked the current guests, "If you're in room N, please move to room 2^N."

Thus, the people in the hotel at that time went to rooms 2, 4, 8, 16, 32, 64, .... Then I went outside and explained my plan to the first few bus drivers, and asked them to pass it on. Here's the plan: each bus received its own prime number, starting at three. So the first bus was 3's, the second 5's, the third 7's, the fourth 11';s and so on, one prime for each bus. Then, as for the people on the bus, they all received powers of those primes. For example, the first bus was assigned rooms 3, 9, 27, 81, 243, and so on, powers of 3 (the N-th person on the bus was assigned room 3^N). The next bus was assigned powers of five, so they had rooms 5, 25, 125, 625, et cetera, the N-th person being assigned room 5^N. There were infinitely many primes, one for each bus, and an infinity of powers of each prime, so everyone had his own room! It took me a while to explain the scheme to everyone--there were a few of them that were math atheists, and it was rough going once or twice, but they all finally settled in.

Then, as I was going over the register, I noticed that no one was in room 6 (= 2x3), nor in room 10 (= 2x5), nor in any room whose number was a product of two or more different primes, since these rooms were not powers of a single prime, and hence had no bus assigned to them. A quick calculation showed that there were, in fact, an infinite number of vacancies! Incredible! I had taken an infinite hotel that was full, added an infinite number of infinities, and when all was done, I still had an infinite number of vacancies!

The point of all this is that infinity is NOT a number, and--though there is a subject called "transfinite arithmetic"--you can't think in terms of doing ordinary arithmetic with infinity. The best way to think about it, is that infinity is a property that some sets possess. Richard Dedekind defined an infinite set to be one which could be put in one-to-one correspondence with a proper subset of itself. It is this strange property that I have played with in the telling of my weird tale.

Incidentally, you might like to know that the hotel closed shortly after that night. Seems there were a lot of lawsuits and stuff, and the last I heard, lawyers -- the number of which is growing without bound -- were convening there. Maybe they'll all be trapped forever, and they won't be bothering common folks any more.

http://scidiv.bcc.ctc.edu/Math/InfiniteHotel.html

Counting To Infinity

The symbol, has been around for more than two thousand years. The Romans used it to represent 1000, a BIG number to them.

About 1650 the English mathematician, John Wallis, proposed that stand for INFINITY, and that stuck.

The concept of infinity has tantalized and sometimes troubled mankind even longer. Zeno of Elea (495 BC?-425 BC?), an early Greek thinker, is remembered for his paradoxes of motion that are rooted in deep questions about the nature of time and space and in some misconceptions about infinity. Most religions attempt to explain in their own ways the mysteries and vagaries of the infinite.

In the early 1600's Galileo began to show signs of a modern attitude toward the infinite, when he proposed that "infinity should obey a different arithmetic than finite numbers." But it was not until the late 19th century that Georg Cantor (1845-1918), a German mathematician, finally put infinity on a firm logical foundation and described a way to do arithmetic with infinite quantities useful to mathematics. His basic definition was simple: a collection is infinite, if some of its parts are as big as the whole. For example, even though from one point of view the entire list of numbers we count with {1,2,3,4,5,.......} is twice as large as the list of even numbers {2,4,6,8,10,.......}, the two lists can be matched-up in a one-to-one fashion.

So the two lists are exactly the same size, infinite. (This idea has been amusingly elaborated in the story of "The Hotel Ad Infinitum" as told by David Stacy.)

Cantor was able to demonstrate that there are different sizes of infinity. The infinity of decimal numbers that are bigger than zero but smaller than one is greater than the infinity of counting numbers. (Click to see Cantor's "diagonalization proof.")

Here's a question that might tax your intuition: There are infinitely many COUNTING NUMBERS and infinitely many FRACTIONS. Which infinity is greater? Think about it, then click for an explanation.

The bottom line is that weird things can happen with infinity.

http://scidiv.bcc.ctc.edu/Math/infinity.html