175, 275 -Homework 10 and suggestions for next week

Homework 10 is due Tuesday, November 18, at the beginning of lecture. The usual considerations apply.

In 175 we will try to cover this week until section 8.7 at least; it is possible this won’t happen until next week. Sections 8.3, 8.4, and 8.5 all introduce important tests for convergence of series; make sure you understand the arguments being presented (rather than just trying to memorize the tests). The material in section 8.6 is particularly important (conditional and absolute convergence). We will also cover some additional material on the -series .

Next week we will continue from section 8.7 (or at whatever point in the chapter we find ourselves at that point) on. We will also cover a few additional topics that the book doesn’t mention or doesn’t treat in sufficient detail, once we are done with 8.10. If you want to read ahead, the topics we will cover are uniform vs. pointwise convergence (including Wierstrass test) and infinite products. I will distribute notes of the topics not covered in the book.

In 275 we will cover from section 13.6 on, but the emphasis will be on section 13.8, and the notion of Jacobian. We will also present a few notions from linear algebra to make sense of the general version of the chain rule. Afterwards, we will continue with Chapter 14, which contains the main results from this course you are likely to use in the future. Chapter 14 will take some time to cover.

Homework 10:

175: Do not use the solutions manual for any of these problems.

Turn in the problems listed for Homework 9 that you still have pending.

Besides the exercises you have pending from last week, there are 15 new problems. Turn in the exercises you have pending, and at least 7 of the new problems. The others (at most ) will be due December 2, together with the additional exercises for that week. 275:

Section 13.5. Exercises 4, 8, 19, 22, 30, 45.

Section 13.6. Exercises 6, 11, 24, 30.

Section 13.7. Exercises 13, 21, 37, 60, 78, 79.

Section 13.8. Exercise 1, 6, 16, 21.

There are 20 exercises this week. Turn in at least 10. The remaining problems (at most 10) will be due together with a few additional exercises on December 2.

Marginalia 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.

Given 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 […]

No, 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 […]

As 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 […]

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.

Let me add something to Noah's nice answer. If there are transitive set models of set theory, then there are such models of $V=L$, and therefore there is a countable $\alpha$ such that $L_\alpha$ is a model (by the Löwenheim–Skolem theorem and condensation). Since $L_\alpha$ is countable, for any forcing poset $\mathbb P\in L_\alpha$ there are (in $L$) […]

The answer depends on the underlying set theory and the actual symbol under consideration, whether $\in$ or $\subseteq$. In standard (ZF) set theory, the axiom of foundation prevents the existence of any set as specified. The reason is that sets have a rank, and the rank of any member of a set $A$ is strictly smaller than that of $A$. However, the rank of po […]

Sure. A large class of examples comes from the partition calculus. A simple result of the kind I have in mind is the following: Any infinite graph contains either a copy of the complete graph on countably many vertices or of the independent graph on countably many vertices. However, if we want to find an uncountable complete or independent graph, it is not e […]

I think that, from a modern point of view, there is a misunderstanding in the position that you suggest in your question. Really, "set theory" should be understood as an umbrella term that covers a whole hierarchy of ZFC-related theories. Perhaps one of the most significant advances in foundations is the identification of the consistency strength h […]

I'll only discuss the first question. As pointed out by Asaf, the argument is not correct, but something interesting can be said anyway. There are a couple of issues. A key problem is with the idea of an "explicitly constructed" set. Indeed, for instance, there are explicitly constructed sets of reals that are uncountable and of size continuum […]