This set is due Monday, September 13.
507 – Homework 1
September 2, 2010
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507: Advanced number theory | 
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Posted by andrescaicedo
A kind of library
Andrés E. Caicedo
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Mathoverflow- Answer by Andrés E. Caicedo for Most memorable titles October 31, 2010Marginalia 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
- Ramsey multiplicity May 26, 2010Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color. For example, $R(3)= 6$, which is usually stated by saying that in a party of 6 people, necessarily there are 3 that know e […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is it consistent to have a function that is sensitive to subset relation from the power set of a set to that set? June 27, 2020No, this is not consistent. Todorčević has shown in ZF that, in fact, there is no function $F\!:\mathcal W(S)\to S$ with the property you require. Here, $\mathcal W(S)$ is the collection of subsets of $S$ that are well-orderable. This is corollary 6 in MR0793235 (87d:03126). Todorčević, Stevo. Partition relations for partially ordered sets. Acta Math. 155 (1 […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Who introduced direct limits? April 19, 2020As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwa […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Atlas-like websites on specific areas of mathematics March 8, 2020A 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.Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Most memorable titles October 31, 2010
- Math.StackExchange
- Answer by Andrés E. Caicedo for Skolem hulls in arbitrary models of some fragments of ZFC November 4, 2020You do not need much to recover the full ultrapower. In fact, the $\Sigma_1$-weak Skolem hull should suffice, where the latter is defined by using not all Skolem functions but only those for $\Sigma_1$-formulas, and not even that, but only those functions defined as follows: given a $\Sigma_1$ formula $\varphi(t,y_1,\dots,y_n)$, let $f_\varphi:{}^nN\to N$ be […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is there an object such that its unique existence follows from choice axiom, while its existence cannot be proven without choice axiom? November 3, 2020I posted this originally as a comment to Alex's answer but, at his suggestion, I am expanding it into a proper answer. This situation actually occurs in practice in infinitary combinatorics: we use the axiom of choice to establish the existence of an object, but its uniqueness then follows without further appeals to choice. I point this out to emphasize […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Series that converge to $\pi$ quickly December 13, 2010I think you may find interesting to browse the webpage of Jon Borwein, which I would call the standard reference for your question. In particular, take a look at the latest version of his talk on "The life of pi" (and its references!), which includes many of the fast converging algorithms and series used in practice for high precision computations […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Reference for consistency strength of ZFC + AD$^{L(R)}$ September 29, 2020The reference you want is MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. Other sources (such as the final chapter of Kanamori's book) briefly discuss the result, but this is the only place where the details are given. More recent papers deal with […]Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Proper and improper forcing Stationarity September 16, 2020For the first question, $\omega_1$ is a subset of $[\omega_1]^{Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Skolem hulls in arbitrary models of some fragments of ZFC November 4, 2020
