Curing the "disease" of transfinite theory is not the simple matter of forbidding speculation on the infinite. It is the challenge to come up with a theory that is richer than transfinite theory. It is the challenge to create a new theory that can preserve the good while cutting out the rot. Personally, I think Georg Cantor and set theorists have made some tremendous strides in our understanding of mathematics. Ultimately we should hope to create a new theory of the infinite that encapsulates these benefits. This essay called Rich Theory is my amateurish stab at the dragon.
The first thing I want to mention is my firm belief that all men and women are capable of reason. Both the classicist and transfinite theorists are guilty of trying to prevent their opponents from speaking. Kronecker's behavior in stifling Cantor's career was a black eye on the integrity of mathematics, just as Hilbert's censoring of LEJ Brouwer was a disgrace of the following generation.
In almost all cases, there is more than one way to prove a given theorem in mathematics. There is not a pure way of thinking. We are all struggling with trying to find our voices, and trying to find the best way to communicate our ideas. The dictatorship of the Bourbaki has been no better for mankind that the dictatorship of the scholastics.
Now, I admit, I have been a little bit unfair to Georg Cantor. He did not develop transfinite theory from idle speculation of the infinite. He was interested in solving a different puzzle. He was interested in defining the continuum. He wanted a way to be able to describe a continuous line as a set of points.
As an avid follower of Immanuel Kant, Cantor embraced the theory of opposites. I believe, he specifically was looking for a way to define the continuum as a catharsis of a thesis and antithesis. Notice how the main goal of the diagonal method is to establish a dichotomy between "denumerable" and "non-denumerable" sets.
Just as Karl Marx was deluded in his view that history was summed up as conflict between the proletariat and bourgeoisie, I fear that Cantor may have been blinded by the theory of opposites and looked specifically for ways to create a thesis and antithesis in explaining the continuum. As a result, I fear that Mr. Cantor may have accidentally introduced false dichotomy into transfinite theory.
This is a matter of human nature: If you look for something, you often find it, even if it isn't there.
(Please note, I too am often guilty of this human defect.)
If we stopped looking for a single dichotomy within infinity, I believe that we will find the study of large and infinite sets to be far richer and more colorful than the black and white study produced Cantor's dichotomy.
Although my muddled mind is not the best tool for undertaking such a study, I have the temerity to make a few suggestions about methodology that a brave soul could use in such an endeavor:
My first suggestion is to drop the dichotomy. Georg Cantor was interested in defining the difference between continuous and discontinuous sets. Let's start our study with an open mind to see what the subject tells us, rather than trying to force our desires on the subject.
In transfinite theory, Georg Cantor tried to extend the finite notions of size and counting to infinite sets. Since it is impossible to count to infinity, the word "denumerable" can only have a counter-intuitive definition when applied to infinity.
Using non-intuitional definitions of the words "counting" and "size" in transfinite theory lead to more confusion than any other part of the theory. Our intuitive understanding of counting is finite and based on our ability to actually complete the process of counting. It is constructive by nature. Redefining the term may make for some good gee-whiz mathematics, but ultimately hurts the aim of mathematical education. Instead, I think we need to use different terminology when speaking of infinite sets.
Its funny, but a lot of our communication problems seem to arise from the limited nature of language. We cannot express more than language permits. I cannot fully describe the effect of a mischievous glance from a young women. Everyone who imagines such an event will have a different face in their mind and assign different meanings to the glance. We have the same limitations in set theory. (We even have these problems when we use symbolic as opposed to phonetic language.)
Although it may be possible for the human mind to conceive of the infinite, language is limited. To make this argument clear, let's simply speak about publishing on the Internet. HTML files, by nature are finite. They begin with an <html> tag. They have a fixed number of characters, and end with an </html> tag. There is only a limited number of bytes on a page.
Even if there are people who can perceive a completed infinity, they are debilitated by the language when they try to communicate the cathartic event of perceiving the whole. Language is finite. We can only communicate the surface of our deeper experiences.
Even if I had the inner eye and conceived the completed infinity, I would still have to bow and admit that I could not transcribe such an event onto paper...even if I chose to use the whole Hebraic alphabet in my scroll.
We have some big limitations in our effort to explain the whole, but the effort may not be totally hopeless. Let's look at what we can do:
I think I started in a positive direction in the section on arbitrarily large sets. In this essay, we found that different operations seemed to cause arbitrarily large sets to grow at different rates. For example, we found that the set of even numbers between zero and n has floor(n/2) members. We found that the set of halves between 0 and n has 2n members. We found that the set of ordered pairs made with the first n integers has n^2 members. The most exciting discovery seems to be that the power set of the first n integers has 2^n members.
The thing we notice about arbitrarily large sets is that different operations seem to create different levels of infinity. For example, lets look at adding five new members to a set with n members (where n is one way big honkin' number). Adding five units to the set is pretty insignificant when compared to the whole. Multiplying the set by 5 is more significant. Squaring the set is on a different level than multiplication, etc.
Do you see what I am hinting at: A rich infinity might preserve these interesting levels that exist in arbitrarily large sets. I believe that if we named these different levels and investigated them fully, we would find some extremely interesting phenomena between aleph-0 and aleph-1. The answers to Hilbert's tenth question might lie in embracing the different levels of infinity.
BTW: It is possible that Georg Cantor was on the right track when he postulated that the continuum existed at the level of the power set of the integers (aleph-1). Accepting a rich infinity does not preclude this possibility. It simply means that n (the size of the set of integers) is on a different level than 2*n (the size of the set of whole numbers) which is on a different level than n^2 (the level of the ordered pairs--which also seems to be the level of dimensionality) which is on a different level than the power set of the integers (2^n).
If you accept the concept of a rich infinity, I think you would find that Cantor's transfinite numbers would behave as they do now...only you have more detail between each of the alephs, and you would have an answer to Hilbert's 10th question: Basically the answer is that there is a ton of cool stuff between aleph-0 and aleph-1. There is multiplication and dimensionality.
Perhaps the most confusing result of transfinite theory is the belief that an infinite set might be the same size as a proper subset of itself. For example, transfinite theorists say that the size of the set of positive even integers is the same size as the set of positive integers. Transfinite theorists say that one to many relations miraculously turn into one to one correspondences at infinity.
Jumping back to the concept of levels, can develop a less confusing terminology. Lets say I call the level of infinity that occurs in power sets the "power level." I could say that the even numbers are of the same "power" as the set of integers. The power set is on a higher level.
If I called the level that occurs at n^2 the dimension level, I could say that the even numbers are of the same dimension as the integers. Hell, for that matter, I could boldly proclaim that the set of even numbers is only the same "power" level as the set of ordered pairs but are of a different dimension.
Since we can construct arbitrarily large sets, I believe that we should be able to construct a consistent theory of levels. My suggestion is that there is a multiplicative level (2n), a dimensional level (n^2) and the power level (2^n). In any case one to many relations stay one to many relations. The size of atoms is not the same size as the set of molecules.
Anyone who read my diatribe about the diagonal method will be surprised to see me ending this discussion on rich theory with a note on the diagonal. It is important to understand that the diatribe was specifically aimed at using the diagonal method to create a dichotomy between the rational and reals. At the end of this lecture, I would like to speak of the real meaning of the diagonal product.
In Two Treatise, Galileo conjectured that the set of squares was the same "size" as the set of integers because you would not run out of squares before running out of integers. There is a one to one correspondence between the sets. From this conjecture, it is easy to conjecture that all infinite sets are the same size.
The diagonal method disproves the Galilean conjecture. It shows that infinity is richer than the single infinity that Galileo envisioned. The fact that we can order the rational numbers in such a way that the diagonal product produces the number 1/3 indicates that the rational numbers is of a different size than the integers. It is just harder to prove.
The diagonal method does not create a dichotomy between the rational and reals. It simply shows that Galileo's conjecture that all infinities are the same size was unfounded. Surely, all infinities share the nature of being boundless, but the diagonal method shows that infinity is not a singularity. In rich theory, I simply suggest that infinity is far richer than the duality proposed by Georg Cantor in his transfinite theory.
Infinity is boundless. Like Ogden Nash's Elephant, there are many different ways to perceive the subject. People like me who rant about people like them rarely do much to improve the world. I simply hope that mathematics can some day overcome the oppositionally based approach to infinity, and provide a richer basis for the study of the transfinite, the continuum and related subjects.
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