June | 2015 | A kind of library

Update

June 29, 2015

So, we moved to Ann Arbor.

We visited the Hands-On Museum (thanks to Zach and Diana, and congratulations!),

and attended the Math Reviews Summer picnic.

I started today.

3 Comments | Life | Tagged: Francisco, Isabel | Permalink
Posted by andrescaicedo

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Mathoverflow
- Answer by Andrés E. Caicedo for Axiom of choice: ultrafilter vs. Vitali set July 6, 2011
  Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter does not imply the existence of a Vitali set. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${ […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Most memorable titles October 31, 2010
  Marginalia to a theorem of Silver (see also this link) by Keith I. Devlin and R. B. Jensen, 1975. A humble title and yet, undoubtedly, one of the most important papers of all time in set theory.
  Andrés E. Caicedo
- Ramsey multiplicity May 26, 2010
  Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color. For example, $R(3)= 6$, which is usually stated by saying that in a party of 6 people, necessarily there are 3 that know e […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is it consistent to have a function that is sensitive to subset relation from the power set of a set to that set? June 27, 2020
  No, this is not consistent. Todorčević has shown in ZF that, in fact, there is no function $F\!:\mathcal W(S)\to S$ with the property you require. Here, $\mathcal W(S)$ is the collection of subsets of $S$ that are well-orderable. This is corollary 6 in MR0793235 (87d:03126). Todorčević, Stevo. Partition relations for partially ordered sets. Acta Math. 155 (1 […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Who introduced direct limits? April 19, 2020
  As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwa […]
  Andrés E. Caicedo
Math.StackExchange
- Answer by Andrés E. Caicedo for Is there an idempotent element in a finite semigroup? April 9, 2013
  When I first saw the question, I remembered there was a proof on MO using Ramsey theory, but couldn't remember how the argument went, so I came up with the following, that I first posted as a comment: A cute proof using Schur's theorem: Fix $a$ in your semigroup $S$, and color $n$ and $m$ with the same color whenever $a^n=a^m$. By Schur's theo […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for MSC of Point Set Topology (Mathematical Subject Classification) February 20, 2021
  It depends on what you are doing. I assume by lower level you really mean high level, or general, or 2-digit class. In that case, 54 is general topology, 26 is real functions, 03 is mathematical logic and foundations. "Point-set topology" most likely refers to the stuff in 54, or to the theory of Baire functions, as in 26A21, or to descriptive set […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Why does my book use $\subseteq$ instead of $\subset$ to describe a transitive set? January 24, 2021
  In the presence of the axiom of foundation, it is true as you indicate that no set belongs to itself, and so the definition of transitive set can be written with $\subset$ (or $\subsetneq$, whichever symbol you prefer). However, one may study also set theories where foundation fails, and then it is natural to define transitive sets in a way that allows self- […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Skolem hulls in arbitrary models of some fragments of ZFC November 4, 2020
  You do not need much to recover the full ultrapower. In fact, the $\Sigma_1$-weak Skolem hull should suffice, where the latter is defined by using not all Skolem functions but only those for $\Sigma_1$-formulas, and not even that, but only those functions defined as follows: given a $\Sigma_1$ formula $\varphi(t,y_1,\dots,y_n)$, let $f_\varphi:{}^nN\to N$ be […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is there an object such that its unique existence follows from choice axiom, while its existence cannot be proven without choice axiom? November 3, 2020
  I posted this originally as a comment to Alex's answer but, at his suggestion, I am expanding it into a proper answer. This situation actually occurs in practice in infinitary combinatorics: we use the axiom of choice to establish the existence of an object, but its uniqueness then follows without further appeals to choice. I point this out to emphasize […]
  Andrés E. Caicedo
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