otto (02/06/83)
* Article ID: ucbvax.721 * From: iheds!ihnp4!houxz!houxi!hou5d!hou5a!hou5e!hou5c!hou5b!hou5f!alice!mhtsa!eagle!harpo!decvax!ucbvax!arens@UCBKIM * Date: Wed Jan 26 19:18:36 1983 * Subject: Re: The Funny Nature of Infinity * Newsgroups: net.math * * From: arens@UCBKIM (Yigal Arens) * Received: from UCBKIM.BERKELEY.ARPA by UCBVAX.BERKELEY.ARPA (3.300 [1/17/83]) * id AA01900; 26 Jan 83 19:18:04 PST (Wed) * To: net-math@ucbvax * * * That's funny? When you finally figure that all infinities are the same, you * realize (or are told...) that there REALLY ARE more real numbers than * rational numbers!! Wait a moment. It is a mistake to say that *all* infinities are the same. Georg Cantor, who began the study of transfinite arithmetic (and was ridiculed for it) identified many different classes of infinity. All the infinities in a given class can have their elements mapped one-to-one onto the elements of any other infinity in the same class. However, it can be shown that this is impossible to do when trying to match up infinities from different classes. The classes are called aleph-sub-0, aleph-sub-1, aleph-sub-2, etc. Aleph-sub-0 contains the infinity of integers, aleph-sub-1 contains the infinity of real numbers, and aleph-sub-2 contains (I believe) the infinity of curves in 2-space. Since it can be shown that a one-to-one mapping can be constructed between the set of integers and the set of rational numbers, this means that aleph-sub-0 also contains the infinity of rational numbers. However, it can be positively shown that *no* such mapping can be made between the set of integers and the set of real numbers, hence the infinity of reals must be in a different class, which is identified as aleph-sub-1. The proof of there not being a one-to-one mapping between the two infinities is really quite simple; it is a reductio ad absurdum proof. For simplicity, we shall proof that no such mapping can even be made between the integers and the subset of reals in the interval (0,1). 1) Assume that there *does* exist a one-to-one mapping between the set of integers and the set of real numbers in the interval (0,1). 2) Then we can list the mapping in a table, showing first the real mapped to the integer 1, next the real mapped to the integer 2, etc., as follows: 1 .Annnnnnn... 2 .nBnnnnnn... 3 .nnCnnnnn... 4 .nnnDnnnn... 5 .nnnnEnnn... 6 .nnnnnFnn... . . . . . . where the n's, A's, B's, etc., represent whatever digits are used by the real numbers in the mapping. Now note the diagonal numbers, represented by the capital letters. From these numbers we can construct a *new* real number in the (0,1) interval that is guaranteed *not* to be in the above table, hence not in the one-to-one mapping. If we do this, we have shown that the table is not really a one-to-one mapping, which is a contradiction, thus showing our primary assumption to be false. How do we find this real not in the mapping? As follows: Let a be a digit different from A, Let b be a digit different from B, Let c be a digit different from C, Let d be a digit different from D, Let e be a digit different from E, and so on. Then we can construct the real number .abcde... formed from all the digits *different* from those on the diagonal shown in the above table, and what's more, this real number is *not* in the table anywhere! It's not in the first position because the first decimal digit of this new number, a, is different from the first decimal digit of the number in the first table position, A. It's not in the second position, because b is different from B. It's not in the third, because c is different from C. It can't be in the N'th position, because our number and the table's number are different at the N'th position. Hence our number is not in the table, hence the one-to-one mapping is not really a one-to-one mapping! QED George Otto Bell Labs, Indian Hill ----------------------