Cantor, Peano, Natural Numbers, and Infinity

Date: 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/