22 | January | 2008 | A kind of library

116b- Homework 2

January 22, 2008

Due Tuesday January 29 at the beginning of lecture.

Update: There is a typo in problem 2, it must be $z<y$ .

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Posted by andrescaicedo

116b- Lecture 5

January 22, 2008

We showed that a function is in $\mathsf{R}$ iff it has a $\Sigma_1$ -graph. It follows that a set is r.e. iff it is $\Sigma_1$ -definable.

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Posted by andrescaicedo

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- Answer by Andrés E. Caicedo for Continuum Hypothesis December 30, 2012
  A classical reference is Hypothèse du Continu by Waclaw Sierpiński (1934), available through the Virtual Library of Science as part of the series Mathematical Monographs of the Institute of Mathematics of the Polish Academy of Sciences. Sierpiński discusses equivalences and consequences. The statements covered include examples from set theory, combinatorics, […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Which journals publish expository work? October 25, 2013
  There is a new journal of the European Mathematical Society that seems perfect for these articles: EMS Surveys in Mathematical Sciences. The description at the link reads: The EMS Surveys in Mathematical Sciences is dedicated to publishing authoritative surveys and high-level expositions in all areas of mathematical sciences. It is a peer-reviewed periodical […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Does "$X \not\to (\omega)^\omega_2$ for every infinite $X$" imply ${\sf AC}$? March 29, 2022
  The answer is no, the statement that for every set $X$ we have $$X\not\to(\omega)^\omega_2$$ does not imply the axiom of choice. This was shown by Kleinberg and Seiferas in 1973, see MR0340025 (49 #4782) Kleinberg, E. M.; Seiferas, J. I. Infinite exponent partition relations and well-ordered choice. J. Symbolic Logic 38 (1973), 299–308. https://doi.org/10.23 […]
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- 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 […]
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Math.StackExchange
- Answer by Andrés E. Caicedo for Are there examples of theorems proved via proper (i.e. non-conservative) extensions? May 7, 2013
  Yes, this is a nice idea, and the approach is used in practice. I list four examples below, but there are many others. Any arithmetic statement, or any first order statement about $(\mathbb R,\mathbb N,+,\times,
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for If a function can only be defined implicitly does it have to be multivalued? June 4, 2011
  Not necessarily. Consider the graph $G$ in ${\mathbb R}^2$ of the points $(x,y)$ such that $$ y^5+16y-32x^3+32x=0. $$ This example comes from the nice book "The implicit function theorem" by Krantz and Parks. Note that this is the graph of a function: Fix $x$, and let $F(y)=y^5+16y-32x^3+32x$. Then $F'(y)=5y^4+16>0$ so $F$ is strictly incre […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Whether the map $x\mapsto x^3$ in a finite field is bijective August 20, 2013
  Following Tomas's suggestion, I am posting this as an answer: I encountered this problem while directing a Master's thesis two years ago, and again (in a different setting) with another thesis last year. I seem to recall that I somehow got to this while reading slides of a talk by Paul Pollack. Anyway, I like to deduce the results asked in the prob […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is it possible to formalize the idea that large cardinal axioms don't limit us, while their negations do? November 25, 2013
  One way we formalize this "limitation" idea is via interpretative power. John Steel describes this approach carefully in several places, so you may want to read what he says, in particular at Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel. Does mathematics need new axioms?, The Bulletin of Symbolic Logic, 6 (4), (2000), 401 […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is $6.12345678910111213141516171819202122\ldots$ transcendental? March 5, 2011
  This is a transcendental number, in fact one of the best known ones, it is $6+$ Champernowne's number. Kurt Mahler was first to show that the number is transcendental, a proof can be found on his "Lectures on Diophantine approximations", available through Project Euclid. The argument (as typical in this area) consists in analyzing the rate at […]
  Andrés E. Caicedo
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116b- Homework 2

116b- Lecture 5

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