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 Atlas-like websites on specific areas of mathematics March 8, 2020
  A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.
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- Answer by Andrés E. Caicedo for Are infinite planar graphs still 4-colorable? December 16, 2013
  As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]
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  Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0
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  Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]
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- Answer by Andrés E. Caicedo for Question about Jech's proof of V = L implies GCH June 3, 2019
  You assume $\omega_\alpha\subseteq M$ and $X\in M$ so that $X$ belongs to the transitive collapse of $M$ (because if $\pi$ is the collapsing map, $\pi(X)=\pi[X]=X$. You assume $|M|=\aleph_\alpha$ so that the transitive collapse of $M$ has size $\aleph_\alpha$. Since you also have that this transitive collapse is of the form $L_\beta$ for some $\beta$, it fol […]
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Math.StackExchange
- 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 What series of books do you recommend before studying the Banach Tarski paradox? February 29, 2020
  I don't think you need too much in terms of prerequisites. An excellent reference is MR3616119. Tomkowicz, Grzegorz(PL-CEG2); Wagon, Stan(1-MACA-NDM). The Banach-Tarski paradox. Second edition. With a foreword by Jan Mycielski. Encyclopedia of Mathematics and its Applications, 163. Cambridge University Press, New York, 2016. xviii+348 pp. ISBN: 978-1-10 […]
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- Answer by Andrés E. Caicedo for How many integer solutions are there to $x+y+z=8$ February 12, 2020
  For the second problem, write $x=-3+x'$ and so on. You have $x'+y'+z'=17$ and $x',\dots$ are nonnegative, a case you know how to solve. You can also solve the first problem this way; now you would set $x=1+x'$, etc.
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- Answer by Andrés E. Caicedo for Cofinality of $2^{ February 10, 2020
  I assume you want to require that $\kappa$ is a limit cardinal; otherwise, if $\kappa=\rho^+$ then $2^{
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