13 | February | 2008 | A kind of library

116b- Homework 5

February 13, 2008

Due Tuesday February 19 at the beginning of lecture.

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Posted by andrescaicedo

We showed that $\mathsf{ACA}_0$ proves the completeness theorem. We then proved the second incompleteness theorem using the result on completeness, the fact that if a model of $\mathsf{ACA}_0$ thinks that it contains a model of $\mathsf{ACA}_0$ then we can really see this latter model, and the diagonal lemma.

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Posted by andrescaicedo

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- Answer by Andrés E. Caicedo for Well known theorems that have not been proved February 3, 2023
  I 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, 2014
  This 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, 2013
  One 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 […]
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- Relative densities September 12, 2013
  Throughout 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 […]
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- On the division paradox December 22, 2022
  This 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
Math.StackExchange
- Answer by Andrés E. Caicedo for Transfinite derivatives December 5, 2013
  In 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 […]
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- Answer by Andrés E. Caicedo for cardinals such that $\alpha < \alpha ^ \beta$ January 31, 2023
  Recall that the beth ($\beth$) numbers are defined by transfinite recursion as $\beth_0=0$, $\beth_{\alpha+1}=2^{\beth_\alpha}$ and $\beth_\lambda=\sup_{\alpha
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- Answer by Andrés E. Caicedo for Prove that Baire space $\omega^\omega$ is completely metrizable? November 30, 2013
  The 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, 2014
  No, 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 […]
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- Answer by Andrés E. Caicedo for How to check the real analyticity of a function? December 3, 2013
  This 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
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116b- Homework 5

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