The goal of my diatribe about the diagonal method is to show legitimate problems that students have with introductory transfinite theory. I wish to encourage thought and debate.
In writing the article, it is my intent to help students understand the controversies involved with transfinite theory. I have intentionally written the articles in a manner that is slightly more provocative than what appears in journals. Again, the goal is to encourage thought.
As my goal is to encourage the reader to think, I have avoided stating my own opinions. Quite frankly, I was really put off in school by professors who basically demanded that I accept this theory without question. I never expect anyone to accept my beliefs without question, in fact I encourage people to question me along with all the other talking heads out there.
Rather than stating my beliefs outright, I took the subtle tact of simply presenting models which only hinted at my ideas. Anyway, some readers have asked that I put up an article that clearly states my position on the method; so that they can hack me to pieces.
Anyway, my two main concerns in the proof lie with what I believe to be a false conclusion in the proof and with foundational issues of logic.
First, I believe that Cantor derived a false conclusion from the diagonal method.
I believe that the primary error in the theory is not with the assertion that the set of Real Numbers is a “different size” than the set of Integers. The problem lies with the assertion that the set of Rational Numbers is the “same size” as the set of Integers. Our finite notion of size just doesn't extend to infinite sets. Putting numbers in a list (i.e., creating a one-to-one correspondence between infinite sets) does not show that they are the same “size.”
This becomes clear if we do a two step version of the diagonal method.
Step One: Lets start with the claim: “Putting an infinite set in a list shows that it is the same size as the set of Integers.”
Step Two: Claiming to have a complete list of reals, Cantor uses the diagonal method to create a real number not yet in the list.
Please, think about this two step model. The diagonal method does not show that the rational numbers are denumerable while the real numbers are not. The diagonal method shows that the assertion in step one is false. The assertion in step one is as false for rational numbers as it is for real numbers.
The diagonal method calls into question the cross-section proof used to show that the rational numbers are the same size as the integers.
My biggest objections with the theory, however, have to do with the foundations of logic. Cantor's work was not simply used as a tool for exploring infinity. It was used as a justification to replace syllogystic logic with a new oppositional logic. The end result of the work in transfinite theory has been to remove the study of basic logic from primary schools.
Cantor's transfinite theory came to light at the height of the new German Idealistic schools. This is the same school of thought that produced Freud's Psychology, Hegel's historicism and Marxist economics. Notice how each of these theories were based on rather strange dichotomies. The German idealists had a strange fascination with secret underlying causes for human action. Marx theories of class conflicts are the best known.
The idea behind oppositional logic is that ideas develop in opposition to each other. Good is defined in contrast with evil, dark in contrast with light. The world spirit evolves through conflicting ideas on the great world stage...Ideas start with a thesis that is then confronted with an antithesis which then culminates in a catharsis. The thinker looks for, then names the great conflicts.
Cantor was interested in the nature of continuity. It would seem natural to apply these new oppositional logics to mathematics. Hence the attraction of the dichotomy between the denumerable and non-denumerable sets.
At the heart of this issue is the very subtle choice of how we choose to describe the attributes of infinite sets. Do we describe them as a dichotomy? or do we describe them simply as attributes of the set?
Although transfinite theory is often introduced as a study of the infinity. The primary concern of the theory is the nature of continuity. What is the difference between a discontinuous set (like the rational numbers) and a supposedly continuous set (like the real numbers)? I tossed in the adverb "supposedly" because neither science nor mathematics has ever completely come up with a perfect theory to explain continuity. When today's physicists explore subatomic phenomenum, they get into a strange world of quantum mechanics.
Anyway, Cantor was interested in continuity. By definition, the rational numbers are denoted by two finite terms. There is a numerator and denominator. The system of real numbers that we use accepts the used of infinite decimals. For example, in decimal notation we write 1/3 as .333... Where the ellipses indicate that string of 3s continue forever. Some numbers such as the square root of two and pi require infinite non-repeating decimals.
Cantor's denumerable/nondenumerable dichotomy is really just a fancy way to say that rational numbers are expressed with finite terms, and real numbers apparently require infinite decimals. Using oppositional logic makes this intuitive description of attributes of a set seem deep and mysterious. It fit in perfectly with the spirit of the day. Hegel had similar conflicts to describe the march of history. Marx has a theory of class struggle, Freud employed a lot of cool terms like ids and egos. Kant, the master of them all, had introduced odd distinctions between a priori, synthetic and a posteriori thought.
The idea of finding a secret dichotomy at the foundations of mathematics that revealed the secret nature of the continuum was a great boon for the proponents of the new dialectic that was sweeping through Germany. With the attention of the academic community focused on subtle arguments about the infinite and dichotomies between the rational and reals, the new thinkers were able to push aside those favoring syllogisms with a new dialectic based on classifying things into sets and oppositional logic...The famous dialectic.
For those who are wondering, the dialectic is the underlying common denominator between Communism and Fascism.
The unfortunate end result of set theory is that basic logic is no longer taught in schools. So while the theory itself appears to be simple idle speculation about infinity, the theory has had real world consequences. The consequence is that we are not giving students the founding in logic that they need to make important decisions in their lives.
The foundational questions are the main reason that I call to question transfinite theory. Should we create a theory based on the opposition of denumerable and non-denumerable sets, or should we develope a mathematical system that simply studies and names the attributes of sets? I favor tossing out the dialectics and using traditional logic.
The terms denumerable, nondenumerable and the attempt to use a one-to-one correspondence on infinite sets is not as productive as simply studying large sets and naming attributes of the sets.
Of course, I admit to my own weakness for philosophical crusades. The real issue, however, should be about what is best for the student. Is teaching that there is a secret dichotomy at the basis of analysis a better way than just teaching students about large sets? Again, I would opt for the straight forward method. My experience and conversations with others tends to indicate that transfinite theory is doing more to confuse the public at large than to help student's develop their ability of critical thinking.
Personally, I believe that Cantor's work has great merit and is a good start to understanding the nature of continuity.
The great controversies surrounding the theory have helped mathematicians create a rigorous foundation for finite set theory. Like most great thinkers, there is a great deal of good in Cantor's work. However, it is not 100% perfect. No theories are. What we need to do now is root out the things that really aren't serving the mathematics community well and expand on those that are working. Keep the wheat and toss out the chaffe.
By removing references to the "size" of infinite sets, I believe that it would be possible to create a new introduction to transfinite theory which preserves the valuable aspects of the theory, while getting us beyond the contention caused by the false dichotomy driven between rational and reals.
By rejecting the dichotomy, I believe that we could create a Rich Theory that does a better job of describing the attributes of large sets.
The Diagonal Method, Roots of Sound Rational Thinking, Links