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© Copyright 2000, Jim Loy
In response to my article on Zeno's Paradoxes, I received an email message saying that Zeno's problem was that infinity was not understood until George Cantor's discovery of transfinite numbers near the beginning of the 20th Century. I have to disagree. While Cantor's work shed important light upon the subject, I think the ancient Greeks showed a very strong and nearly complete knowledge of infinity.
Even Zeno, in his paradoxes, dealt with infinity very well. He just reached erroneous conclusions (maybe he knew they were erroneous, and was just trying to stretch our minds). He based his paradoxes on the erroneous idea that infinitely many steps cannot be accomplished in a finite amount of time.
Euclid dealt with infinitely long straight lines, and showed that they do not do anything particularly bizarre at great distances. For example, assuming that his parallel postulate is true, then parallel lines are equidistant from each other, even way out there toward infinity. He also showed that there are infinitely many primes (See The Infinitude Of Primes). He used regular polygons with infinitely many sides to deal with the circumference and area of a circle. He dealt with infinity in completely natural and confident and very modern ways.
Archimedes extended Euclid's use of infinity by doing infinite sums (to estimate (pi)), thus anticipating integral calculus by about 1900 years. He also anticipated derivatives by about 1900 years. Both of these ideas implied the use of limits, although Archimedes did not seem to be aware of this. Limits are the way in which we deal with infinity, today (since Newton and Leibnitz). We do not deal directly with infinity, we deal with quantities which grow without bounds (approach infinity as a limit). We do not divide by zero, we deal with quantities which get closer and closer to zero (approach zero as a limit). We deal with infinity by never really dealing directly with infinity. So Newton and Leibnitz extended our understanding of infinity by giving us limits, an automatic way of being careful to avoid paradoxes.
Cantor extended our knowledge of infinity, by studying infinite sets. To determine if two lists (sets) contain the same number of elements, we match up the elements. 1,2,3,4,5 and 8,3,22,99,6 both have five elements, as we can match them up like this: 1&8, 2&3, 3&22, 4&99, 5&6. We then move on to infinite sets, like the set (collection or list) of all integers. Surprisingly, there are exactly the same number of even numbers as there are integers (even numbers and odd numbers). This may seem strange, but it is easy to prove because you can list them and match them up 1&2, 2&4, 3&6, 4&8, 5&10, etc. This goes on forever, and there are none left over from either list. See The Hotel Infinity. In a similar way, we can show that there are exactly the same number of rational numbers (fractions, both proper and improper) as there are integers. They can be matched up exactly. Cantor showed that there are many more real numbers (rational numbers and the irrational numbers, like square roots, between the rational numbers) than there are integers. In other words, he showed that they cannot be matched up in any way. So there are two kinds of infinity, the number of integers, and the number of real numbers. And he showed that there are infinitely many kinds of infinity. These kinds of infinity are called transfinite numbers.
There are still mysteries about infinity. But from a practical point of view, we understand infinity very well, as did the ancient Greeks.
On the inside cover of the book The Mystery of the Aleph, by Amir D. Aczel, which is a biography of Georg Cantor, we read the following: "...There are as many points on a line one inch long as on a line a mile long... According to Cantor, infinite sets are equal." Those two sets are equal. But infinite sets are not necessarily equal, as mentioned above. I assume the book does better than the inside cover does.
There are infinitely many positive integers. This is a rather basic idea, but it is based upon an even more basic idea: There is no largest integer (or other number). And so, I can challenge you to name a candidate for the largest number prize, and then I can name a larger one (one more than yours, perhaps). That works for any number. And so, you don't have to puzzle over a vague distant concept like infinity. You can deduce it from the more easily understood concept, there is no largest number.
Mathematicians often cringe at the expression, "infinite number." Infinity is not a number. And so, I avoid saying "infinite number" or "equals infinity." Instead I say "infinitely many" or "is infinite." See Two Equals One?, which deals with infinity, in an educational way.
You may have noticed an infinity setting on the focus ring of your camera. You can actually focus on infinity with a camera lens (or with a telescope). This is really infinity; the camera lens is focused at a point infinitely far away (parallel light rays). But, for all practical purposes, closer objects focus just fine, as if they were out there at infinity. That may help you see that infinity is not something mysterious, but is a real limit that some of us must deal with. You could actually focus beyond infinity, if the camera would only let you turn the ring just a little farther. You can easily do that with a telescope.
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