This is a follow up of a previous post where a formalization of the concept of behavior –bisimilarity– was presented. Now I want to focus into one approach to characterize “equivalence of behavior,” that is, when two states of a system are bisimilar.
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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.
¡Empezamos este miércoles!
Este miércoles a las 14hs, comenzará el curso Teoría de Conjuntos, dictado por su servidor en la Universidad Nacional de Córdoba.
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Modelos del Universo
Después de reflexionar un rato sobre las pruebas de independencia en Teoría de Conjuntos que usan el método de “forzamiento” o forcing, una conclusión que se puede sacar es que no es tan anti-intuitivo poder agregar un conjunto nuevo, i.e., que no se pueda obtener a partir de los ya existentes usando las operaciones usualmente aceptadas de definición de conjuntos. Por ejemplo, no nos produce ninguna inquietud tomar un anillo arbitrario $k$ y adjuntarle un elemento trascendente sobre él, obteniendo $k[x]$.
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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):
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).
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.
Happy $\binom{2^6}{2}$!
In the Southern hemisphere we are enjoying summer games, like drawing some Boolean algebras.
Boolean algebra with 6 atoms
If you’re bored, you may try and check if the drawing is correct.
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.
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Teoría de conjuntos ingenua
Abstract. I’ll discuss one view of the naïve-axiomatic dichotomy in Set Theory. My claim is that one leaves the “naïve” world when first order logic (or put differently, the possibility of different models of ZFC) becomes explicit.
En muchas ocasiones se utiliza el término teoría de conjuntos ingenua; incluso el libro de Halmos se llama así.
Anti-Elección
El Axioma de Elección (AC, por sus siglas en inglés) dice que dada una familia $\calF$ de conjuntos no vacíos, puedo elegir un elemento de cada uno. Hay varias maneras equivalentes de formular esto más precisamente:
- Hay una función $f$ tal que $f(A)\in A$ para todo $A\in\calF$;
- (si los conjuntos de $\calF$ son disjuntos) Hay un conjunto $C$ tal que su intersección con cada $A\in\calF$ es un singulete;
- etcétera.