I received a letter on Friday from Bob Kustra, the president of BSU.
“I am pleased to inform you that you are being promoted to the rank of Associate Professor and being awarded tenure.”
A kind of library
Andrés E. Caicedo
Recent Comments
Mathoverflow- Answer by Andres Caicedo for How to rearrange only negative part of a conditionally convergent series to get any sum greater then initial? November 29, 2010Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]Andres Caicedo
- Answer by Andres Caicedo for What is the consistency strength of a standard model of ZF versus a transitive model? December 27, 2014What precisely do you mean by a standard model? An $\omega$-model? (That is, a model whose set of natural numbers is isomorphic to $\omega$.) Or a $\beta$-model? (That is, a model whose ordinals are well-ordered.) If the latter, the Mostowski collapse theorem tells us any such model is isomorphic in a unique way to a unique transitive model. If the former, t […]Andres Caicedo
- Answer by Andres Caicedo for Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper? December 23, 2014This is Theorem 39 in the paper (see Theorem 4.(i) for a user-friendly preview). But the fact that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$ is older (1946) and due to Erdős, see here: Paul Erdős. Some set-theoretical properties of graphs, Univ. Nac. Tucumán. Revista A. 3 (1942), 363-367 MR5,151d. (Anyway, it is probably easier to read a more modern presentation, […]Andres Caicedo
- Answer by Andres Caicedo for Failure of diamond at large cardinals July 18, 2013This is a nice problem. Here is what I know. (Below, I refer to the Handbook. This is the Handbook of Set Theory, Foreman, Kanamori, eds., Springer, 2010.) First of all, the consistency of the failure of diamond at a weakly compact cardinal seems open. Woodin has asked this explicitly, I do not know if the question itself is due to him. Of course, $\diamonds […]Andres Caicedo
- Answer by Andres Caicedo for Proofs of the uncountability of the reals. November 23, 2010I thought about this question a while ago, while teaching a topics course. Since one can easily check that $${}|{\mathbb R}|=|{\mathcal P}({\mathbb N})|$$ by a direct construction that does not involve diagonalization, the question can be restated as: Is there a proof of Cantor's theorem that ${}|X|Andres Caicedo
- Answer by Andres Caicedo for How to rearrange only negative part of a conditionally convergent series to get any sum greater then initial? November 29, 2010
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- Answer by Andres Caicedo for Uncountable linearly independet family in $K^\mathbb{N}$ February 16, 2015An example I like is this: Say two sets of natural numbers are almost disjoint iff they are infinite but their intersection is finite. Suppose $\mathcal F$ is an uncountable family of almost disjoint sets. Then the collection of characteristic functions of the sets in $\mathcal F$ is an example. (If $A$ is a subset of $\mathbb N$, then its characteristic fun […]Andres Caicedo
- Answer by Andres Caicedo for Is $L_{\omega+1}$ uncountable? February 9, 2015The set $\mathrm{Def}(X)$ consists of all subsets of $X$ first-order definable in the structure $(X,\in)$ from parameters. What matters here is that there are only countably many formulas, and only $|X^{Andres Caicedo
- Answer by Andres Caicedo for Is there a series where the terms tend to zero faster than the harmonic series but it still diverges? December 1, 2013There is no slowest divergent series. Let me take this to mean that given any sequence $a_n$ of positive numbers converging to zero whose series diverges, there is a sequence $b_n$ that converges to zero faster and the series also diverges, where "faster" means that $\lim b_n/a_n=0$. In fact, given any sequences of positive numbers $(a_{1,n}), (a_{ […]Andres Caicedo
- Answer by Andres Caicedo for Is a dense and co-dense subset $G_\delta$ or co-$G_\delta$ January 27, 2015Yes. In fact, by a counting argument, most dense co-dense sets are neither $G_\delta$ nor $F_\sigma$. The point is that there are exactly as many $G_\delta$ or $F_\sigma$ as there are real numbers, but there are as many dense co-dense sets as there are sets of real numbers. In somewhat more detail: There are only countably many rationals, so there are counta […]Andres Caicedo
- Answer by Andres Caicedo for Uses of ordinals January 14, 2015There are quite a few examples in analysis, typically through the use of transfinite induction or transfinite recursion of length $\omega_1$. The Baire-Cantor stationary principle states that if $C_0\supseteq C_1\supseteq C_2\supseteq\dots\supseteq C_\alpha\supseteq\dots$, $\alphaAndres Caicedo
- Answer by Andres Caicedo for Uncountable linearly independet family in $K^\mathbb{N}$ February 16, 2015
Congratulations! Here’s to a transfinity of good research in the future!
Thanks!
It is a little late, but congratulations!
Not late at all. Many thanks!
Congratulations!!!
Thank you, John!
Congratulations, Andres!
Many thanks, Joel!