And so we reach the end of yet another diatribe on transfinite theory. It seems to just go on and on and on, doesn't it? Well, I hope my ranting was informative and entertaining. We learned several things:
- We learned that the proof really is about a completed infinity--not just a run of the mill potential infinity.
- We learned that the set of namable numbers is both denumerable and nondenumerable.
- We learned that that, when performed on finite binary strings, the diagonal method simply proves that 2n > n for n > 1. (Not really an earth shattering discovering).
- We found ways to twist the diagonal method to show that particular orderings of the rational numbers are incomplete.
- We also found that we can put the power set of the integers in a table. A set that is supposed to be denumerable.
I didn't spend any time on the all important self-referential paradoxes that troubled Bertrand Russell. The goal of this article was not to be a complete treatment of the paradoxes of transfinite theory, but to make the argument that the standard introduction to transfinite theory is insufficient for establishing the Cantorian dichotomy between the rational and the reals.
Transfinite theory, as it currently exists has too many pot holes and detours. I will throw my suggestions about how to fix the problem in the essay Rich Theory. Rich Theory contains suggestions like eliminating the use of the term denumerable. Instead of talking about the "size" of infinite sets, it talks about different levels of infinity. Rich theory builds on arbitrarily large sets, and tries to avoid invoking actual infinities.
Personally, I find myself more and more enamored with the classical mathematicians. Constructive mathematics is much more useful than transfinite theory. I agree whole heartedly that it is best to stay with mathematics that you can perform on a computer. Anyway, if you wish to read further, I have included a few additional essays:
In its present state, transfinite theory does more harm than good. I hope that someday, a brave soul takes on the academicians, and presents us with a better defined, more intuitive description of infinity. As for now, I will leave with an update on our friend Brother Mathematicus.
Brother Mathematicus dipped his pen in a well of the finest Indian Ink. Carefully, he scrolled on a parchment the aleph. The aleph, the most wonderous of all symbols, preserved by the Gods to serve mankind as the first of the transfinite numbers.
Aleph-one follows aleph-naught...
The initiates gathered in the classroom, as our guardian of the transfinite mysteries prepared for the presentation of the rites. Brother Mathematicus had been performing the rites for over a half century. He saw but two of initiates in that time bloom into publishing set theoreticians. The rest had simply faded into the great nothingness of mediocrity.
The class of initiates seemed smaller this year. The students seemed more contentious. It is the computers he thought. Computer savvy students seemed to have a different notion of logic, and large numbers. They are less interested in ideas that they cannot program into their digital devices. Constructionist, his heart stuttered, the students were one by one falling into the camp of Kronecker: Leopald Kronecker, the anti-christ, the blasphemer, the persecutor of Cantor.
Brother Mathematicus gazed from his chamber toward the class. Three of the students had laptops with wireless modems. The new students not only failed to humble themselves before the alter, they flaunted their indifference. Brother Mathematicus looked at the students computers. They were playing some sort of multiuser battle game...
Yea, they play Doom on the steps of the temple!
Brother Mathematicus faltered at the thought. He closed his eyes and recited the three steps of the diagonal method. He recited the rites of the one-to-one correspondence.
One-to-many relations becometh one-to-one as sets reacheth infinity.
He recited the scared method for putting the rational numbers in a table:
Tabulation yields denumeration.
Finally he invoked the sacred diagonal itself. He closed his eyes as he tried to envision the diagonal of the completed infinity. He saw diagonal digits transforming from zeros to ones and ones to zeros. He felt a burning in his mind as the inner eye began to open--The finiteness of his corporal existence opening to the divinity and oneness of the eternity. He felt the inner eye begin to awake...
...then emptiness, absolute emptiness as the illusion fell short of his ultimate desires.
Brother Mathematicus knew of the inner eye. He knew that there was an inner eye possessed by the greatest of the great. He knew that less than one half of one percent of the people could possess such a miracle, and see infinity.
He had dreams of that wonderful completed infinity--the infinity where the rational numbers became the same size as the integers. It must be a vision of pure beauty and delight. He longed for the day that the inner eye would awaken, and he could see the glorious vision.
Loath to admit it, Brother Mathematicus, held to his bosom the vilest of all confession: Brother Mathematicus, guardian of the great temple of set theory, was mired in the same pestilence as the common man. Like the beasts in the fields, and pigs in the style, commoners wallowed in complete ignorance...never to enjoy the subtleties of mathematics. Yet here stood the keeper of the gate, himself unable to enjoy the elation of oneness. The great inner eye lay dormant in his soul.
Brother Mathematicus would recite the incantations of the diagonal method. He would raise his hands as he recited the dichotomy between the rationals and reals. But, he, would not partake in the rapture of the students who saw the many become one. He would not see the dichotomy rip between the rationals and reals--the dichotomy that created the whole of mathematics. Brother Mathematicus was but the humble messenger. The inner eye lay blind in his soul.
-- the horror.
History of Infinity : Curing
The Disease: A Rich Theory:
Descriptive Mathematics, Index, Sponsors
© 2002 Kevin Delaney