Abstract. In this series of posts I’ll discuss problems that can be posed in an elementary way but the only way to solve them (to the best of my knowledge) is to develop some set theory. This post is dedicated to a problem appearing in Fraenkel’s Set Theory , that states that you can change the iso type of any total order by adding just one point. The solution depends on well orders, which I consider a part of the theory of sets.
Leyendo diversas fuentes, encontré dos problemas elementales cuya solución involucra desarrollar algo de Teoría de Conjuntos “seria”. En este post plantearé uno de ellos.
Cómo romper un orden total
Un orden total es un conjunto $L$ con una relación “$<$” irreflexiva, transitiva, y para la cual vale tricotomía: se da alguna de $x<y$, $x = y$ ó $x>y$ para cualquier par $x,y$ en $L$.
Academic life has given me a lesson for the second time, and it was harsh this time.
The first lesson had a happy ending, and it can be described by the following words:
Beware of the paper bin!
This happened on 2008. I was experimenting with a really wonderful software, the bundle Prover9-Mace4, by the late Bill McCune. My aim was to obtain a family of examples of equational classes where some formula grows unboundedly in complexity (this formula was related to direct product representations). Continue reading
Assume you want to prove a Theorem X. If you’re a fan of reductio ad absurdum (RAA), you start by saying “Assume that Theorem X is false. Hence…” and after some reasoning, you reach a contradiction. You write as a closing sentence, “This contradiction shows that Theorem X must be true.”
I want to argue about the following questions: Do we need only one use of the rule of contradiction? Can we start a proof as above and not use an argument by contradiction but at the end? Continue reading