Refuting the Diagonal Method

The goal of the first iteration of this critique of the diagonal method was simply to encourage those who have questions about the diagonal method to think. Many pop math texts would list the diagonal method or the infinite hotel and leave a confused audience.

I've been giving thought of late of moving from parody to actually trying to write a clearer refutation of the diagonal method and the denumerable/nondenumerable dichotomy at the base of transfinite theory.

Unfortunately, refuting Transfinite Theory is as difficult a task as refuting its kissing cousins: Marxism, Fascism, Nazism and all the other impressive isms built on the dialectics of the German Transcendental Idealistic movements.

Transfinite theory is a nasty piece of work that combines paradoxes and oppositional dialectics to create a dichotomy to explain the mysterious difference between discrete and continuous mathematics.

The theory starts by using Bolzano's interpretation of Galileo's paradox to infer that the set of natural numbers is the same "size" as the set of rational numbers. The second step of the method uses the diagonal method, a form of the liar's paradox, to suggest that the real numbers are different size. This is the denumerable/nondenumerable dichotomy.

The dichotomy essentially states that at n = ∞, n = n^2 while n != 2^n.

Transfinite theory is essentially a dichotomy couched between two paradoxes. I admit, my distaste for the theory is based mainly on the fact that the method injects paradoxes and oppositional dialectics into the very foundations of mathematics.

Yet, it is a near impossible task to refute systems based on paradoxes as it is the paradoxical nature of the theory its adherents enjoy. See, once you have paradoxes built into the foundations of a system, it is possible to simply switch the interpretation of the paradoxes as the situation dictates.

While it is possible to refute a scientific theory simply by demonstrating that the scientific theory doesn't match reality. Disproving a metaphysical theory based on paradoxes must involve convincing people that such theories are less desireable than open, clearly stated methodologies that can be refuted.

It is a tough game. When trying to refute a system based on paradoxes, you never know in advance what twists the dialecticians will take to defend their system. There is no true proof or disproof of the theory because the theories make no claims that can be tested or verified.

In effect, a refutation of a paradoxical system must fight with one arm tied behind its back. The refutation must clearly state its objectives. While the defenders of transfinite theory can twist their arguments and definitions until any holes created by the refutation get patched.

Fortunately, with the growing influence of computers, people have rediscovered the wonders of discrete logic. With computers, people are more aware of the power and dangers of recursive function, etc..

Meanwhile, on the political front, the Berlin Wall fell. There is a widespread recognition of the failures of Transfinite Theories of Communism and Fascism. There is even a small number of voices in the academic community that acknowledge the attrocities that occurred under the various regimes that adopted the dialectics of the transcendental idealism.

There is hope that as the other offsprings of Kant's transcendental idealism fall into disrepute that the same fate might befall transfinite theory. Rather than just creating a refutation of the theory, the best approach to bringing an end to system of dichotomies couched in paradoxes is to create mechanisms that can describe sets that are not paradox bound.

Unfortunately, I am not a very good mathematician, and developing such a replacement system is beyond my means. I called my attempt to create an alternative to the denumerable/nondenumerable dichotomy Rich Theory. Rich Theory is based on the observation that arbitrarily large sets seem to develop different layering patterns. Rich theory seems to salvage the parts of transfinite theory that are most valuable, while pushing the paradoxes to the side.

Anyway, below you will find a first cut at trying to develop a refutation for the denumerable/non-denumerable dichotomy. The cut is extremely rough. I admit, this is not my favorite subject, and I want to spend time on work I actually care about. I think Poincare was correct in his analogy that this dichotomy couched in paradoxes is a disease, and it is just a matter of waiting until the academic community tires of their toy.

The Refutation

NOTE: I started writing this refutation after reading Everything and More by David Foster Wallace. Mr. Wallace does a great job presenting the view that many transfinite theorists hold of history.

Many popular math texts introduce their ideas in the form of a historical perspective. The first several pages of this new version of the critique looks at the history of the development of logic and paradoxes. These sections are intended for readers interested in the historical philosophical aspects of the theory. Readers interested only in the math can skip these sections and read the Galileo's paradox and diagonal method.

The new four part outline is as follows. (You can also still read the contents of the orginal diatribe below, with I list below this outline.)

The Ancients:

In exploring the history of the ancient Greeks, I wish to emphasize that there are some foundational theories that encourage and promote discourse. Other theories seem to stop discourse dead in its tracks. When the Ancient Greeks were faced with paradoxes of the infinite, they chose to emphasize finite logic. By differentiating between the actual and potential infinity they were able to make use of infinity in their investigations, but were able to avoid the problems associated with the paradoxes.

Galileo's Paradox

In this article, I explore the nature of Galileo's paradox and how there are multiple mappings between infinite sets. Galileo's paradox occurs because different mappings have different attributes.

The Dialectics:

Transfinite Theory is one of the many foundational systems based on Kant's "transcendental idealism" and Hegelian metaphysics. This section questions the wisdom of basing mathematics on such a slippery slope.

The Diagonal Method

This is the meat of the presentation. This article has several goals:

In this article, I present the diagonal method. I show that the diagonal method is a clever reworking of the liar's paradox. I then take to task Cantor's position that the diagonal method drives a dichotomy between the rationals and reals. This article explores the diagonal method on finite binary strings and shows that the set of namable numbers is both denumerable and non-denumerable.

Rich Theory

In Rich Theory, I wish to show that removing the dichotomy between rational and reals, that we can develop a very interesting layering of the infinite. The diagonal method was simply a short cut Cantor developed in his efforts to explore the nature of continuity. This work is not dependent on the oppositional logic used in the denumerable/non-denumerable dichotomy that is supported in traditional presentations of the diagonal method.

Original Presentation

Since its inception toward the end of the nineteenth century, transfinite theory has been a center of great controversy. Many mathematicians see it as a paradise–the finest example of abstract mathematics. Others see it as a disease waiting a cure. Many schools present the conjecture as if it were fact. The theory is both the source and justification of new math. Even worse, transfinite theory has been one of the primary motivating factors for removing traditional logic (with Aristotelian roots) from the classroom and replacing it with the dialectics of the new Kantian/Hegelian tradition.

From the vantage of descriptive mathematics, I find myself siding with those rejecting the theory. The standard presentation of transfinite theory is marred with paradoxes and confusing terms. For example, it is physically impossible to "count" an infinite set, yet we classifly infinite sets as "countable" and "uncountable."

The ideal of Descriptive Mathematics is that we use mathematics to communicate our ideas. The extremely confusing use of terms in the standard introduction to transfinite theory does the opposite. It hinders our ability to communicate ideas. It injects into the foundations of mathematics incomprehensibles and twisted definitions that boggle the mind and, in most cases, simply add to the epidemic of math anxiety.

The main hope of this article is that mathematics professors will actively read this article and look for ways to improve the introduction to transfinite theory so that it is more conducive to learning and growth of a student, and less of an exercise in the black art of cleverness.

This article is in two parts: In the first part, I refute the standard presentation of the diagonal method. In the second part, I suggest an alternative method for discussing the infinite. This second method starts with an exploration of arbitrarily large sets and the power set.

Unfortunately, it is almost impossible to talk about transfinite theory without coming across as a madman. Even the biographers of Georg Cantor seem to question the man's sanity. Likewise, mathematicians such as Henri Poincare and LEJ Brouer have had their formidable contributions to science sullied simply by trying to address this difficult issue.

Since it is impossible to discuss the theory without coming off as a diatribe, I accepted the inevitable and will try to make this work an entertaining diatribe. Although I sympathize with the intuitionists and others who described transfinite theory as a disease, I also believe that there is a great deal of value to Cantor's work, and it is worth the effort to correct the terminology and fix the gaping holes in the theory. I dislike the present state of mathematics where transfinite theory creates a great deal of rancor and feeds math anxiety.

In the final essay, Rich Theory, I will start to piece together my answer to the problem. My goal is not simply to "cure" transfinite theory, but trying to find ways to enrich our understanding of large sets.

Anyway, infinity has always been a area full of mad ranting. So please, sit back and enjoy and enjoy my brief diatribe.

The Diatribe

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start 1/12/2002 ©Kevin Delaney