07 | April | 2010 | A kind of library

187 – Handout

April 7, 2010

If you need today’s handout on the pigeonhole principle and g.c.d.s, let me know and I’ll bring extra copies to lecture.

The exercises in the handout are extra-credit. All extra-credit problems, other than the two due this Friday, are due the last day of lecture. (I may change this date, if need be, but let’s try it for now.)

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Mathoverflow
- On the structure of maximal Ramsey colorings October 25, 2021
  For positive integers $a_1,\dots,a_n$, recall that the multicolor Ramsey number $R(a_1,\dots,a_n)$ is the smallest integer $N$ such that if the edges of the complete graph $K_N$ are colored with the $n$ colors $1,\dots,n$, then there is some $i\le n$ and a set of $a_i$ vertices, all of whose edges received color $i$. A maximal Ramsey$(a_1,\dots,a_n)$-colorin […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for How to rearrange only negative part of a conditionally convergent series to get any sum greater then initial? November 29, 2010
  Georgii: 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 […]
  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, 2011
  It 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 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
Math.StackExchange
- Answer by Andrés E. Caicedo for Why isn't the equal sign part of the set theory language? August 12, 2021
  Equality is part of the background (first-order) logic, so it is included, but there is no need to mention it. The situation is the same in many other theories. If you want to work in a language without equality, on the other hand, then this is mentioned explicitly. It is true that from extensionality (and logical axioms), one can prove that two sets are equ […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Well-ordering of the reals in ZF with constructibility? April 19, 2018
  $L$ has such a nice canonical structure that one can use it to define a global well-ordering. That is, there is a formula $\phi(u,v)$ that (provably in $\mathsf{ZF}$) well-orders all of $L$, so that its restriction to any specific set $A$ in $L$ is a set well-ordering of $A$. The well-ordering $\varphi$ you are asking about can be obtained as the restriction […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Incompleteness in reverse July 7, 2021
  Gödel sentences are by construction $\Pi^0_1$ statements, that is, they have the form "for all $n$ ...", where ... is a recursive statement (think "a statement that a computer can decide"). For instance, the typical Gödel sentence for a system $T$ coming from the second incompleteness theorem says that "for all $n$ that code a proof […]
  Andrés E. Caicedo
- 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
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