Aside

Existen diferentes tamaños de infinito

English version below

Debo hacer muchísimas tareas, pero es imposible evitar demorarme diez minutos en esto. Y lamento de antemano que probablemente sólo sean profesionales de la matemática quienes lean estas breves líneas.

(Cantor, ca. 1874) No todos los conjuntos infinitos son iguales. Hay diversidad de infinitos, en tanto número. En cantidad, hay más puntos en una recta que números naturales.

Hoy, en el colectivo hacia mi trabajo, venía corrigiendo typos de mi apunte de Teoría de Conjuntos. Un jovencito sintió curiosidad y me preguntó qué era. Le expliqué que era un curso de posgrado, también para últimos años de Licenciatura, y a modo ilustrativo, le comenté el resultado de Cantor. Obviamente, no lo conocía, pero lo entendió instantáneamente. Lo contrasté con el hecho que hay igualdad de número entre los puntos de la recta y los del plano, como también entre los naturales y los enteros.

Luego de casi siglo y medio, esto es una pieza de la cultura universal. Hay que universalizarlo, y esto trasciende mi natural apología por la Teoría de Conjuntos. Regálenlo a todos los que quieran. Continue reading

Models of the Universe

I promised my friend Zoltán a translation of an older post in Spanish, Modelos del Universo, so here it is. It is almost a literal version.


After some reflection on independence proofs in Set Theory using the method of forcing, one of my conclusions was that it should not be too counterintuitive the fact of adding a new set, i.e., a set not obtainable from previous existing ones using the already established operations of set construction.  For instance, we don’t feel any remorse for taking an arbitrary ring $k$ and adjoining an element transcendental over it, obtaining $k[x]$. Continue reading

Behavior

I’ll review in this post one of the most important notions of equivalence of behavior used in Computer Science: Bisimilarity.

Actually, “behavior” is a very deep word, and it is likely that one can not give a precise mathematical definition of what it means. But in a restricted context, there’s such definition and its surprising degree of simplicity is an indication of how fundamental it is.

Continue reading

Image

Introducing Modley

I have a brand-new favicon at this blog. It is actually a very simple, albeit special, smiley (or “emoji,” of you prefer):

Modley

The first model-theoretic smiley!

Modley is, as most of you have already noticed, the “consequence” or “satisfaction” symbol used in model theory and in Logic in general, rotated 90° counterclockwise. You may typeset the actual symbol $\models$ in $\LaTeX$ by using \models in math mode.

One can metaphorically say that the sole introduction of this symbol signals the beginning of modern mathematical logic, since it works as the bounds between mathematical structures (on the left) and the language in which we write their properties (on the right).

Video

Obscene Mathematics

Several years ago, I had the idea of setting up a mathematics website. One candidate name was “F!cktorial. Mathematics under consent of the King.” Luckily, I was unable to attain this goal, but I found a YouTube username, Numberphile, that almost fits the bill.

I’ve watched two of Numberphile’s videos, they were rather interesting. The first one is about a formula that “plots itself”.

The second video concerns some rather strange ways we name numbers.

Naïve set theory

This post is sort of a translation and a follow-up of a post in Spanish about the comparison between naïve and axiomatic set theory.

The point I made in the previous post is that

One leaves naïve set theory in the moment that first order logic (FOL) gets explicit.

Or, from a different perspective, when you realize in full the possibility of different models of set theory.
Continue reading