Cantor, Peano, Natural Numbers, and InfinityDate: 03/19/98 at 10:08:20 From: David Bennett Subject: Please help me! Sorry for the dramatic subject line. For the last four or five years, I have been wrestling with a math problem that I cannot summarize in less than about four or five typed pages. It deals with transfinite numbers, specifically with contradictions I believe I have spotted in Cantor's original paper introducing the diagonal method. Even for the English teacher that I am, I am not at a particularly high level of mathematical accomplishment, but I do enjoy reading popularizations of mathematics, some of which have been among my finest reading experiences. It has gotten to the point that I can not read any layman's math books, certainly none on set theory or the mathematics of the infinite--they just send me off into the same arguments I have conducted with myself time and again. It is an obsession. I need to find someone who can read my little paper and persuade me that I am wrong--if I am shown the error, I will be able to admit it, and I know it must obviously exist. As an American living in Japan, I cannot by any means I know of contact any math professor in my area; can you read my paper or connect me to someone who will? I need someone who knows set theory and Cantor's transfinite mathematics. The paper is lucid and not idiotic. I am the eagerest possible student, just waiting for a teacher who can spare the time (not inconsiderable, I know) to read four or five pages. Can you hook me up with someone? David Bennett Date: 03/20/98 at 08:11:23 From: Doctor Jerry Subject: Re: please help me! Hi David, This is not a responsive answer to your request, rather an explanation as to why it is not highly probable that you will receive a responsive answer. I suppose there are two reasons. First, we are able to respond to only 30 to 40 percent of the questions we receive. This is a matter of available man- or woman-power. Most of us are volunteers. Each of us scans questions as they come in, judging by the subject line if we want to look at the message itself. If it's not in our area of expertise, we probably skip the message altogether. Second, (and here I'm taking a somewhat extreme case) if the message is from someone who believes he (usually it's a male) has trisected a general angle and, moreover, tends to believe that the mathematicians of the world are in a conspiracy to suppress any contributions from non- professionals, we mostly skip such messages. It is my opinion that most of us believe that short of putting that person through a standard math curriculum, so that he will be operating under the same rules for proof that we take for granted, it is largely a waste of time to try to persuade the person of his error. We believe we know that it has been proved that trisection of a general angle with straight edge and compass is not possible. Coming back to your concerns, most of us are not experts in set theory and logic, but have our training in algebra, analysis, number theory, or ... But most of us believe that that ground has been gone over by many, many very bright individuals. As a matter of faith and training, I suppose, we would tend to be skeptical of claims that there are contradictions in Cantor's original paper. Well, perhaps you might get a response, if only from me, if you were to state in a paragraph or two, the substance of one of the contradictions you have found. -Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 03/20/98 at 09:55:08 From: David Bennett Subject: attn: Dr. Jerry I'm the one who wrote about the "contradictions" I believe myself to have found in Cantor's diagonal method (I use the quotes to acknowledge the absurdity of what I am saying). I don't know if I can summarize my "attack" on Cantor briefly enough, but you might be able to answer a question that arises from my ideas: is there a consensus in the mathematics community as to whether infinitely large natural numbers exist? Cantor, in his original paper, which I don't have in front of me, says something like "the set of FINITE (my emphasis) natural numbers." And then says no more about it. However, consider this attribute of natural numbers: In any subset of the natural numbers, the value of the greatest natural in the subset must be equal to or greater than the number of members of the set. If you start counting from 1, then the number of members of whatever subset you choose is equal to the greatest member. In the subset {1, 2, 3} the number of members is 3, and the greatest member is 3. If you skip any naturals, then the greatest member will be greater than the number of members of your subset. In the subset {1, 2, 7}, the number of members is 3, but the greatest member is 7. It is without doubt true that no human can show me a set that violates my rule. But Cantor wants us to believe that there are an infinite number of members of the set of naturals (otherwise there's no point in talking about them in this context), the greatest member of which is merely finite, thus violating my rule, which seems to be a valid characteristic of natural numbers. Re-read this paragraph, and think about it--can any set of naturals have a number of members greater than the number representing the greatest member? It is not enough to say that when you reach the realm of infinity, different rules apply--infinity is not a "realm," infinity is the mundane fact of applying the same rules over and over again. If you stop applying those rules, then you've stopped, and to stop is not infinity. Besides, I've given a good rule--if you're going to stop using my rule, you must at least give a reason. Cantor does not--that's why I ask if there is a pre-existing mathematical convention against infinitely large naturals. Here's another argument for infinitely large naturals. Consider any real number between 0 and 1, the set that Cantor, more or less, treated in his paper. Now take that real number, which is a string of digits extending out to the right of the decimal point. Using the decimal point as an axis of rotation, "flip" the string 180 degrees, if you will. For example pi - 3 = 0.14159.... when "flipped" equals ...95141.0, with the ellipsis (the "...") out to the left instead of the right. This typographical operation is every bit as possible as Cantor's diagonalization, which is also a typographical algorithm. Having "flipped" this infinitely long string representing a real, seeing now a string extending out infinitely far to the left (as reals do to the right) I ask you, what does this string of digits represent if not a natural? If there are infinitely large naturals, then they are represented by infinitely long strings, as are reals. If there is at least one infinitely large natural then, like all naturals, it has an infinite number of successors, giving us an infinite number of infinitely long strings, allowing us to diagonalize over this set just as well as we can over the set of reals. This would totally overturn Cantor's proof. Therefore Cantor must deny infinitely large naturals, which he very quietly does. But this seems to violate the rule I gave above about the number of members of a set of naturals compared to the greatest member of that set, and this seems an extremely solid rule. I've skipped over a lot, and I've omitted one of the highlights of my argument (what I believe to be the type error that arises from the "set" of natural numbers--you'll have to live without that one). From what I have said so far, perhaps you notice that I believe that Cantor has limited the set of naturals to the POTENTIALLY infinite, while allowing the set of reals the status of the ACTUALLY infinite. Thus he has built his conclusion (two different infinities) into his premises. This is highly condensed, and as I said, I've omitted one of my best arguments. What do you see as a flaw in what I've given you? I realize (as a teacher myself--students come up with all sorts of questions) that you might not be able to address my particular points, but if not, can you tell me if you know whether or not there is a mathematical convention that forbids infinitely large naturals, and if there is, whether it addresses the matter I have raised concerning number of members and value of the greatest member in sets of naturals? I know this will be hard to answer, but I want you to know that I appreciate your response to my first inquiry, and I appreciate your attention to this. I've written the authors of various books I've read through their publishing companies with no response from anyone. I have not yet stooped to just sending a form letter to as many professors as I can get the address of, but I'm going to do that, I suppose. I really want to be truly persuaded that I am wrong (since I don't think it highly probable that I'm right, and I need to dump this whole set of ideas out of my beleaguered mind), but in order to do so, I've got to get a response. Perhaps I can claim to be different from those who have trisected an angle in that I know I'm probably wrong, but it's just no good until I SEE that I am wrong. But I admit I am like those people in that I can't help believing myself to be right (I just can't see it!) Anyway, you're the first mathematician who has responded to me in any way whatsoever, and I hope you realize that says something about you. David Bennett Date: 03/21/98 at 08:27:22 From: Doctor Jerry Subject: Re: attn: Dr. Jerry Hi David, There are those (called Platonists) who believe that all mathematical concepts and objects exist and are, as it were, waiting in the wings to be discovered. Others believe that mathematics is the creation of persons. For most of us, these are not burning issues. However, our "general" graduate education in mathematics tends towards the latter viewpoint, at least operationally. During my first year in graduate school, I vivdly remember starting with Peano's postulates for the natural numbers and, over the course of two or three weeks, constructing the non-negative integers, the rationals, the real numbers, and the set of complex numbers. The idea was that Peano's Postulates postulated the existence of a set with certain properties. We didn't have to believe that such a set exists in any concrete sense. Of course, these things resembled, by design, the "whole numbers' we had been working with since pre-kindergarten. Here are Peano's Postualates: 1. Let S be a set such that for each x in S there is a unique x' in S. 2. There is an element in S, we shall call it 1, such that for every element x in S, 1 is not x'. 3. If x and y are in S such that x' = y', then x = y. 4. If M is a subset of S such that 1 is in M, and, for every element x of M, x' is also in M, then M = S. The x' operation is supposed to suggest adding 1 to x (but, of course, at this stage addition is not a defined operation). Postulate 4 is the idea behind the method of proof called mathematical induction. One quickly proves that addition satisfies the usual properties: associative, commutative, and distributive laws. For notation, we agree that 2 = 1', 3 = 2', etc. Here are several comments, in order of their appearance in your note: 1. I don't know why Cantor said something like "the set of finite natural numbers." I know people who say, for emphasis, "the set of finite real numbers." This is redundant language for most of us. He did construct what are called infinite cardinal numbers and infinite ordinal numbers (an entire, separate field), but I've never heard about infinite natural numbers. I have a hunch that he was just adding some emphasis or was making some reference to infinite cardinal numbers. Cantor preceded Peano in the development of mathematics. 2. You said: In any subset of the natural numbers, the value of the greatest natural in the the subset must be equal to or greater than the number of members of the set. Within the mathematics I know, this statement is flawed. Why? Because for many subsets of N, there is no "greatest natural." Here's an example (of course, I'm assuming that N satisfies Peano's Postulates): Let S = {2,4,6,8,...}, that is, the set of the even numbers. If e were the greatest even number, then, by the meaning of "greatest," there could be no even numbers larger than e. Yet, (e')' would be an even number (and so, a member of S) larger than e. This is a contradiction. If you add one word to your statement, making it: In any FINITE subset of the natural numbers..., then the statement is true. Indeed, one can prove from Peano's Postulates that any finite subset S of N has a largest member, call it e. It will, of course, be larger than the the number of elements of S (since some elements less than e may not be in S). 3. It's harder to comment on "flipping" 0.14159... I think the difference between this operation and Cantor's diagonal process is that he gives a clear recipe or procedure for determining the first diagonal element, and then the second, and, in principle, any specific diagonal element. I say "in principle" not to hide something, but to admit that if someone wanted me to calculate the 10^{10^{10}} th digit of pi, I could imagine doing it, but, in practice, couldn't produce the digit. You would have to say what the first digit of the flip of 0.14159... is and then what the second digit is, etc. Otherwise, no one would agree that you have a recognizable object. Oh, it is something in the world of ideas, but, operationally, it is not a natural number, a real number, or a member of any set I've heard about. -Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 03/21/98 at 14:48:08 From: David Bennett Subject: attn:Dr. Jerry--last one I know I'm overstaying my welcome with this, but I won't bother you again after this one. Please bear with me for a couple more arguments. What diagonalization does is prove the trivial fact that there are more than n n-digit numbers. In Cantor, what makes it non-trivial is that n is infinity, not to put too fine a point on it. Now, let's say I'm going to prove that there are more than four 4-digit numbers. First make a list of four of them: 1111 1112 1113 2225 Now, I add one to each digit as I diagonalize (the nth digit of the nth member as I descend the list), and get 2226, which is different, and that's fine. I've generated (specified a precise name for) a number proven different from the original list, thus proving that list incomplete, qed. The point is, I can't stop until I reach the 4th digit of the 4th string. If I do the first 3 members of my list, then I've got 222... and I don't yet know if I'm actually generating a "new number," as I call it, until I reach that last member and diagonalize over that last digit. So when Cantor, in his thought-experiment, diagonalizes over an infinitely long list of infinitely long lists of digits (real numbers, in the relevant interpretation), he's got to quite literally REACH that last digit in order for diagonalization to work. To deny, as you did, or rather Peano, that there is a greatest member of the set of even numbers, is to deny that Cantor can reach this digit, don't you think? The last digit of an infinite list of digits is equivalent to the last number in an infinite list of evens. To deny that such "last" things exist is to believe in potential infinity. This is what Cantor does for the naturals. But to reach that digit--necessary for Cantor's proof--is to believe in actual infinity, as he does for the reals. That's what I mean by proving two levels of infinity by already using two different standards for infinity in his assumptions, making his proof circular. Also, when you speak of my flipped numbers as being, let us say, impractical in any application, I would say that Cantor's generated number is not only equally impractical, but far less well-defined. Cantor's number is purely theoretical--by definition, it is a number that no human can ever lay eyes on. If anyone had ever seen that number, surely it would have been included in the original list over which Cantor diagonalizes! It is a pure thought experiment, without any possibility of even being started in reality. Whereas pi - 3, flipped, is every bit as clearly specified as pi is. On Cantor's unseen, unimaginable, and certainly not very useful number, rests an entire field of mathematics. But I I wanted to be brief. These matters are for fun, and by God, math can be fun, no matter what they say. If anyone ever asks you (I saw a lot of questions on Cantor at the Dr. Math site), Rudy Rucker's book _Infinity and The Mind_ really explains all this stuff in a surprising amount of detail. Not all his books are all that great, but this one is a real treat for laypeople interested in infinity. I'll leave you in peace now, and look back over my Peano. David Bennett Date: 03/23/98 at 08:01:23 From: Doctor Jerry Subject: Re: attn:Dr. Jerry--last one Hi David, >What diagonalization does is prove the trivial fact that there are >more than n n-digit numbers. In Cantor, what makes it non-trivial is The standard belief is that Cantor proved that the number of real numbers cannot be enumerated with the natural numbers, that is, you can't tie white tags numbered 1,2,3,... to the real numbers. You can do it, but you'll run out of tags long before you run out of real numbers. What Cantor did was to suppose that you could tag the real numbers. Then, using the tagged numbers, which presumably included every real number (actually, I think he dealt with just part of the real numbers, those between 0 and 1), constructed a number that doesn't appear among those tagged. This is a contradiction. Hence, the reals can't be tagged in this way. Cantor named the cardinality of the natural numbers aleph sub 0. He showed that the cardinality of the set of reals is bigger than aleph sub 0. The cardinal numbers are a new creation. The finite cardinals are "isomorphic" to the natural numbers but the cardinals go on past the finite cardinals. >To deny, as you >did, or rather Peano, that there is a greatest member of the set of >even numbers, is to deny that Cantor can reach this digit, don't you >think? No, I don't think this. The proof I gave, together with my intuition, convinces me that there is no greatest member of the set of even numbers. For me, this is independent of Cantor's proof; it is (no insult intended) just not relevant. >the last number in an infinite list of evens. To deny that such >"last" things exist is to believe in potential infinity. This is what >Cantor does for the naturals. But to reach that digit--necessary for >Cantor's proof--is to believe in actual infinity, as he does for the >reals. That's what I mean by proving two levels of infinity by >already using two different standards for infinity in his >assumptions, making his proof circular. I have an uneasy feeling that you are mixing two kinds of infinity. While it is true that the set of natural numbers is not a finite set, that doesn't imply that there must be natural numbers that are infinite. The number of finite numbers is infinite. >Also, when you speak of my flipped numbers as being, let us say, >impractical in any application, I would say that Cantor's generated >number is not only equally impractical, but far less well-defined. What I intended to say was that while your flipped numbers are a conceivable object, they are not part of the natural numbers or real numbers. All of us are free to create, at least in thought, various entities, but whether they fit into previously created objects is another question. The set of natural numbers has certain properties; how would you add 2 and the flip of 0.159... Is this operation commutative, associative, etc.? To prove these things would be your job; most of us would not accept your assertion that it just must be so. Okay, enough. If you like to read books about mathematics or mathematicians, I can recommend G. H. Hardy's _A Mathematician's Apology_. The writing is lucid and interesting. -Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ |
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