Due Tuesday February 26 at the beginning of lecture.
116b- Homework 6
February 20, 2008
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116b: Computability and incompleteness | 
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A kind of library
Andrés E. Caicedo
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Mathoverflow- Answer by Andrés E. Caicedo for Well known theorems that have not been proved February 3, 2023I am not sure whether this qualifies as "well-known". Anyway, in set theory, in the study of the partition calculus (transfinite generalizations of Ramsey's theorem), effort centered for a while in studying relations of the form $$ \omega^m\to(\omega^n,k)^2 $$ for $m,n,k$ positive integers. Here, exponentiation is in the ordinal sense. This re […]Andrés E. Caicedo
- Answer by Andrés E. 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 MR0009444 (5,151d). (Anyway, it is probably easier to read a more modern pre […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for The Erdős-Turán conjecture or the Erdős' conjecture? June 3, 2013One of the best places to track these things down is The mathematical coloring book, by Alexander Soifer, Springer 2009. Chapter 35 is on "Monochromatic arithmetic progressions", and section 35.4, "Paul Erdős’s Favorite Conjecture", is on the problem you ask about. As far as I can tell, the question is sometimes called the Erdős-Turán con […]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
- On the division paradox December 22, 2022This question is partly motivated by Timothy Chow's recent question on the division paradox. Say that a set $X$ admits a paradoxical partition if and only if there is an equivalence relation $\sim$ on $X$ such that $|X|Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Well known theorems that have not been proved February 3, 2023
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- Answer by Andrés E. Caicedo for Transfinite derivatives December 5, 2013In Ralph P. Boas's A primer of real functions, page 118, this is discussed in the following way: The derivative of infinite order of $f$ is defined on an interval $I$ iff the sequence $(f^{(n)})$ converges uniformly on $I$ (it is enough to require uniform convergence on compact subsets of $I$). Call $L$ the limit of this sequence, so $L$ is continuous a […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for cardinals such that $\alpha < \alpha ^ \beta$ January 31, 2023Recall that the beth ($\beth$) numbers are defined by transfinite recursion as $\beth_0=0$, $\beth_{\alpha+1}=2^{\beth_\alpha}$ and $\beth_\lambda=\sup_{\alphaAndrés E. Caicedo
- Answer by Andrés E. Caicedo for Prove that Baire space $\omega^\omega$ is completely metrizable? November 30, 2013The point here is that two functions are close iff they agree on an initial segment, that is, $d(f,g)\le 1/(n+1)$ iff $f(0)=g(0),f(1)=g(1),\dots,f(n-1)=g(n-1)$. Now, if $(f_n)_n$ is a Cauchy sequence, then, for each $n$, there is $N_n$ such that for all $m,k>N_n$ we have $d(f_m,f_k)\le1/(n+1)$. That is, all functions $f_m$ with $m>N_n$ agree on their f […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius? January 7, 2014No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for How to check the real analyticity of a function? December 3, 2013This is a difficult question in general. Ideally, to show that $f$ is analytic at the origin, you show that in a suitable neighborhood of $0$, the error of the $n$-th Taylor polynomial approaches $0$ as $n\to\infty$. For example, for $f(x)=\sin(x)$, any derivative of $f(x)$ is one of $\sin(x)$, $\cos(x)$, $-\sin(x)$, or $-\cos(x)$, and the error given by the […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Transfinite derivatives December 5, 2013
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