One-to-One Correspondence and Transfinite NumbersDate: 12/10/1999 at 00:00:41 From: Lisa Huang Subject: Set Theory Hi, I am doing a project in which I have to explain the set theory that George Cantor discovered in the late 1800s. I got an article from the Math Forum site that explained how he went about it, but it has math terms that I don't really understand. I don't understand what he means by putting elements into "one-to-one" correspondences, and I also don't really understand what a "transfinite number" is. The page that I went to is: http://mathforum.org/~isaac/problems/cantor2.html If you could please explain just the very, very basics of set theory to me, I would be very grateful. Thank you for your time. Date: 12/10/1999 at 19:04:50 From: Doctor Ian Subject: Re: Set Theory Hi Lisa, Let's say that you have a collection of baseball cards, and a collection of pens, and you want to know which collection is larger. Normally, you would count the number of cards, count the number of pens, and compare the two numbers. But what if you didn't know how to count? Or you knew how to count, but only up to 5 or so? Well, you could start pairing cards with pens: take one card and one pen, put them off to the side; take another card and another pen, put them off to the side; and so on. If you run out of cards first, you know that you have more pens, while if you run out of pens first, you know that you have more cards - even though you can't say exactly how many you have of either. And if you run out of both cards and pens at the same time, then you know you have the same number of each. In this case, what you've done is to put the cards and the pens into a one-to-one correspondence. This is sort of the situation that Cantor was in when he wanted to show that the number of integers is the same as the number of rational numbers. Obviously he couldn't just count all the integers and all the rationals and just compare the numbers. But what he _could_ do was try to set up a one-to-one correspondence between the two sets of numbers. In effect, he wanted to pair up each integer with one particular rational number. Then he could do the same thing with integers and rational numbers that we were just doing a moment ago with baseball cards and pens. But it's a tough problem, because between any two rationals, you can always put another rational. So if he tried to do something like this, 1 <-> 1/2 2 <-> 1/3 3 <-> 1/4 and so on, someone could always say, "But you've left out all the rationals between 1/2 and 1/3," or "You've left out all the rationals between 1/3 and 1/4," and so on. So it looked as if he would always run out of integers first, which, to be truthful, is what 'common sense' says should happen. However, by laying the rationals out in a checkerboard pattern, : : : 1/4 2/4 3/4 ... 1/3 2/3 3/3 ... 1/2 2/2 3/2 ... 1/1 2/1 3/1 ... he found that since every rational would show up somewhere on the board (do you see why?), and since every square on the board could be assigned a unique integer, 10 11 12 ... ----------+ 5 6 7 | ------+ | 2 3 | 8 | --+ | | 1 | 4 | 9 | he had in fact found a one-to-one correspondence between integers and rationals. So, how many rationals are there? The number of rationals is the same as the number of integers. They are both infinite. But there are other sets that are larger than either of these sets - for example, the set of real numbers. So while it's tempting to just use one word - 'infinity' - to describe the sizes of all these sets, that would cause confusion. So he decided to give different names to the different kinds of infinities, and those names are what we call 'transfinite numbers.' I hope this helps. Set theory is a big topic so I'm not going to try to explain it all to you in this message. But be sure to write back if you have questions about specific things you're having trouble understanding. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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