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.”
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A kind of library
Andrés E. Caicedo
Recent Comments
Mathoverflow- Answer by Andrés E. Caicedo for Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$ May 10, 2018The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Does constructing non-measurable sets require the axiom of choice? October 14, 2010As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]Andrés E. Caicedo
- Relative Densities September 12, 2013Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Solutions to the Continuum Hypothesis May 18, 2010(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Are there any non-linear solutions of Cauchy's equation ($f(x+y)=f(x)+f(y)$) without assuming the Axiom of Choice? March 6, 2011It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$ May 10, 2018
- Math.StackExchange
- Answer by Andrés E. Caicedo for Set of discontinuity of monotone function is countable May 23, 2018The usual definition of a series of nonnegative terms is as the supremum of the sums over finite subsets of the index set, $$\sum_{i\in I} x_i=\sup\biggl\{\sum_{j\in J}x_j:J\subseteq I\mbox{ is finite}\biggr\}.$$ (Note this definition does not quite work in general for series of positive and negative terms.) The point then is that is $a< xAndrés E. Caicedo
- Answer by Andrés E. Caicedo for Hadwiger–Nelson problem 5 colors needed if color sets are Lesbeauge measureable November 11, 2017The result was proved by Kenneth J. Falconer. The reference is MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189. The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue m […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Can there be an $\aleph_2$ or even greater? May 7, 2018Yes, there is an $\aleph_2$ and an $\aleph_3$, and there are alephs beyond all the $\aleph_n$. A suitable version of Cantor's diagonal proof is perfectly general and shows that, for any set $X$, $|X|Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Showing in $ZFC-Foundation$ that $V\neq R$ May 1, 2018Given a class $S$, to say that it can be proper means that it is consistent (with the axioms under consideration) that $S$ is a proper class, that is, there is a model $M$ of these axioms such that the interpretation $S^M$ of $S$ in $M$ is a proper class in the sense of $M$. It does not mean that $S$ is always a proper class. In fact, it could also be consis […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Constructible universe and the standard model April 30, 2018As the other answers point out, the question is imprecise because of its use of the undefined notion of "the standard model" of set theory. Indeed, if I were to encounter this phrase, I would think of two possible interpretations: The author actually meant "the minimal standard model of set theory", that is, $L_\Omega$ where $\Omega$ is e […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Set of discontinuity of monotone function is countable May 23, 2018
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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!