Unas palabras en torno al conjunto de todos los conjuntos

I. Si se acepta la existencia del conjunto $\mathbf{U}$ de todos los conjuntos entonces del axioma de separación (t.c.c. Aussonderungsaxiom o esquema de compresión) se sigue que también tendría sentido hablar del conjunto $\mathbf{E}:=\{x \in \mathbf{U}: x \notin x\}$. ¡Y es precisamente este conjunto $\mathbf{E}$ el que origina las situaciones raras que (históricamente) motivaron la revisión del tratamiento intuitivo de la teoría de conjuntos! Específicamente, $\mathbf{E}$ es un conjunto tal que no es posible decidir si pertenece o no a sí mismo. Por un lado, $\mathbf{E}$ no puede ser elemento de sí mismo porque la propiedad que cumplen los elementos de $\mathbf{E}$ es la de no ser elementos de sí mismos. Por otra parte, si $\mathbf{E}$ no es elemento de sí mismo entonces $\mathbf{E}$ satisface la propiedad que define a sus elementos y, por lo tanto, debería ser el caso que $\mathbf{E} \in \mathbf{E}$...

II. Otra manera de caer en la cuenta de los detalles que surgen al aceptar la noción del conjunto de todos los conjuntos es como sigue (el argumento lo aprendí hace algún tiempo de Andrés Caicedo en MSE): Cantor probó mediante su método diagonal el resultado (ahora clásico) que asegura que no es posible dar una función suprayectiva que vaya de un conjunto $\mathrm{X}$ a su conjunto potencia ($2^{\mathrm{X}}, \mathcal{P}(\mathrm{X})$, etc.: cada quien que siga la notación que más sea de su agrado, de gustibus non est disputandum); por otro lado, si $\mathbf{U}$ denota, como en el párrafo anterior, al conjunto de todos los conjuntos entonces la función $f: \mathbf{U} \to 2^{\mathbf{U}}$ dada por $t \mapsto t$ sería suprayectiva. ¡Contradicción!

Claramente, podríamos seguir con los debrayes sobre estas cuestiones pero prefiero retomarlos en otro momento. El tema de los conjuntos es medular en matemáticas. Incluso soy de la opinión de que sobre la noción de conjunto (en matemáticas) también se podría decir lo que Borges apuntara sobre el infinito en La perpetua carrera de Aquiles y la tortuga: "... [es una] palabra (y después concepto) de zozobra que hemos engendrado con temeridad y que una vez consentida en un pensamiento, estalla y lo mata".

Auf wiedersehen!

El reporte de Coxeter sobre el caso de Matusalén

H. S. M. Coxeter hacía notar en su artículo An ancient tragedy (Math. Gaz., 55 393 (1971), pág. 312) lo siguiente:

«En Génesis 5: 25-29 se lee que:

"... Vivió Matusalén ciento ochenta y siete años, y engendró a Lamec. Y vivió Matusalén, después que engendró a Lamec, setecientos ochenta y dos años... Fueron, pues, todos los días de Matusalén novecientos sesenta y nueve años; y murió. Vivió Lamec ciento ochenta y dos años, y engendró un hijo; y llamó su nombre Noé...".

Por otro lado, en Génesis 7: 6 se lee esto:

"Era Noé de seiscientos años cuando el diluvio de las aguas vino sobre la tierra...".

Por consiguiente, Matusalén murió en el año del diluvio pues en tal año su edad era de

187 + 182 + 600 = 969.

años...»

Después de revisar la cuentas, es posible que algún lector por ahí se pregunte: ¿Murió Matusalén en el diluvio? O en otras palabras, ¿permitió Noé que su abuelo pereciera en el diluvio?

En The magic numbers of Dr. Matrix (Prometheus Books, NY, 1985, pág. 176.), Martin Gardner menciona que, de acuerdo al rabino Shlomo Yitzchaki (1040-1105), la respuesta a esta cuestión es negativa: según Yitzchaki, Matusalén murió unos días antes del inicio del diluvio; Dios envió el diluvio una vez que pasaron los siete días de duelo por el fallecimiento de Matusalén. Al parecer, Yitzchaki basaba su hipótesis en Génesis 7: 4 ("Porque pasados aún siete días, yo haré llover sobre la tierra cuarenta días y cuarenta noches...").

Una anécdota retomada del "muro" de Ken Ribet

«There's a long story about this page, and I might as well tell it here. I uploaded this image because one of my facebook friends asked me about it yesterday [9/11/2009]; she had heard mention of it at the MSRI, but the story that she heard was—how shall I put it—not completely accurate.

Best to tell it as I remember it...

I took a graduate algebra course at Brown when I was an undergraduate. I bought a copy of Lang's "Algebra (3rd edition)" around that time. It was the bible of graduate algebra, and I suppose that it still is. When I was at Harvard, I studied very hard for the written qualifying exam that we sat for at the end of our first year of graduate work. I spent months mastering the core material in algebra, topology and analysis. I got a little frustrated by Lang's style and wrote the comment that you see in black ink. Years later, Serge had my office in Princeton while I was away in Paris. As usual, he consulted his own books, and he found my comment. He added his own (bottom of the page) and told me about it when I returned to the US. We both had a good laugh.

I was amazed, years later, when a German mathematician asked me about this page when we were at the Canadian number theory meeting in Vancouver—this must have been 20 years ago. How did the guy know? Serge or I must have talked, and word had gotten around.

I made this image in July, 2001 by pointing my first digital camera at the page and pressing the shutter. Serge was in town, and I told him that I wanted to link this image to the web page for the graduate algebra course (Math 250A) that I was about to teach. That way the students could see it. "Of course!" As the semester progressed, mathematicians around the world found out about the link on my web page and began forwarding the URL to their friends. One day Serge phoned me in my office: "You mean anyone in the world can see this? Take it down!" So I did.»

An oft-cited comment of Mr. Darwin

"During the three years which I spent at Cambridge my time was wasted, as far as the academical studies were concerned, as completely as at Edinburgh and at school. I attempted mathematics, and even went during the summer of 1828 with a private tutor (a very dull man) to Barmouth, but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense..."

Fuente: The Autobiography of Charles Darwin... Me enteré del origen de esta cita a través de un artículo reciente de D. H. Bailey y J. M. Borwein en la sección de ciencia de The Huffington Post (Why E. O. Wilson is wrong... [Fecha de publicación: 04/17/2013, 4:34 P.M.]). Al momento que escribo esto, lo único que encuentro con respecto a esta cita en la entrada en inglés de Wikiquote sobre Charles Darwin es lo siguiente:

Mathematics seems to endow one with something like a new sense.

• Attributed without source to Darwin by E. T. Bell, Men of Mathematics (1937), page 16.
• However Bell is well known to have mythologized in his writing.
• This statement does not appear in Darwin Online or in the Darwin Correspondence Project.

Dedicamos esta entrada a todas aquellas personas para las que la precisión en este tipo de asuntos es esencial...

I. Tamm and the remainder term in Taylor's theorem

George Gamow, the physicist... who escaped to the United States from Stalinist Russia, tells the following tale of what can befall an innocent scholar in times of political turbulence.

Here is a story told to me by one of my friends who was at that time a young professor of physics in Odessa. His name was Igor Tamm (Nobel Prize Laureate in Physics, 1958). Once when he arrived in a neighbouring village, at the period when Odessa was occupied by the Reds, and was negotiating with a villager as to how many chickens he could get for half a dozen silver spoons, the village was captured by one of the Makhno bands, who were roaming the country, harassing the Reds. Seeing his city clothes (or what was left of them), the capturers brought him to the Ataman, a bearded fellow in a tall black fur hat with machine-gun cartridge ribbons crossed on his broad chest and a couple of hand grenades hanging on the belt.

'You son-of-a-bitch, you Communistic agitator, undermining our mother Ukraine! The punishment is death'.

'But no', answered Tamm. 'I am a professor at the University of Odessa and have come here only to get some food.'

'Rubbish!', retorted the leader. 'What kind of professor are you?'

'I teach mathematics'.

'Mathematics?', said the Ataman. 'All right! Then give me an estimate of the error one makes by cutting off Maclaurin's series at the n^th term. Do this and you will go free. Fail and you will be shot!'

Tamm could not believe his ears, since this problem belongs to a rather special branch of higher mathematics. With a shaking hand, and under the muzzle of the gun, he managed to work out the solution and handed it to the Ataman.

'Correct!', said the Ataman. 'Now I see that you really are a professor. Go home!'

Who was this man? No one will ever know. If he was not killed later on, he may well be lecturing now on higher mathematics in some Ukrainian university.

W. Gratzer, Eurekas and euphorias: the Oxford book of scientific anecdotes. Oxford University Press 2002, p. 36.

De Siegel a Mordell

Göttingen, March 3, 1964.

Dear Professor Mordell,

Thank you for the copy of your review of Lang's book. When I first saw this book, about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible. My feeling is very well expressed when you mention Rip van Winkle!

The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory—Lagrange, Gauss or, on a smaller scale, Hardy, Landau. Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one. I see a pig broken into a beautiful garden and rooting up all flowers and trees.

Unfortunately there are many "fellow-travelers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched. These people remind me of the impudent behaviour of the national socialists who sang: "Wir werden weiter marschieren, bis alles in Scherben zerfällt!"

I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction—I call it: Theory of the empty set—cannot be blocked up. Let us hope that your review may be helpful.

I still remember the nice time we had together during your visit in Göttingen.

With best wishes, also to Mrs. Mordell,

Carl Siegel.

[De acuerdo con lo que se puede leer en la cuenta de David C. Kandathil en Google+, el original de esta carta se encuentra en la biblioteca del College of St. John de la Universidad de Cambridge.]

Apuntes para una historia del problema de los NPI

«... In 1985, a high school senior in Hawaii named David Williamson, who was taking mathematics courses at the University of Honolulu, proved that if an odd perfect number exists, it must have exactly one prime factor that, when divided by 4, leaves a remainder of 1. Williamson's professor didn't know whether the result was original and suggested he write to the legendary Erdös, who had just come through town. Williamson eventually got a letter back from Erdös: 'The result you proved is in fact due to Euler. He also proved that every even perfect number is of the form 2^p-1(2^p-1) where 2^p-1 is a prime. It is also known (proved by Carl Pomerance) that an odd perfect number if it exists must have at least 7 distinct prime factors. Perhaps the following problem of mine will interest you...' Williamson, now a combinatorialist at IBM, was thrilled. 'This letter to a high school student won't rank very high on Erdös's list of accomplishments, but it did mean a lot to me.'»

Paul Hoffman, The man who loved only numbers: the story of Paul Erdös and the search for mathematical truth. Hyperion, NY, 1998, pp. 47-48.