But let us remember that we are dealing with infinities and indivisibles, both of which transcend our finite understanding...
This is the introduction to the first iteration of my critique of the diagonal method. When I wrote the first iteration of the article, I simply wished to make a little fun of the meta lords who dictate from ivory towers. The third iteration (A Tale of Two Paradoxes has a more complete version of my views on the subject.
Those readers wishing to learn the party line should read David Foster Wallace's new work Everything and More,
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.
Many of the problems with transfinite theory lie with the diagonal method and the definition of the term "denumerable."
- A Parody of the Diagonal Method
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This parody is frighteningly similar to my introduction to transfinite theory. Yes, the professor dimmed the lights, included disparaging remarks about the common man, and came up with the statistic that only one half of one percent of people can see the completed infinity--arrrggghhhh!!!!
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- A Review of the Diagonal Method
A brief review of the diagonal method.
- The Diagonal Method on Finite Sets
When performed on finite sets, the diagonal method simply shows that number of permutations of a string of characters is greater than the length of the string. That is 2^n > n, where n > 1.
- One-to-One Relations
- Paradoxes (Russell's Paradox on the Web)
- Set of Namable Numbers
- The Relative Size of Arbitrarily Large Sets.
- Power Sets
- A Clarification of My Own View (added 10/17/2003)
- An Opinionated History of the Infinity: My look at the history of Infinity.
- Al's Infinite Hotel (added 5/13/2003) Exams the flaws in the infinite hotel model.
- Curing the Disease: Notes on the difficulty of curing the disease.
- A Rich Theory: This is my suggestion for creating a new, richer definition of the infinite.
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