175, 275 -Homework 9 and suggestions for next week

Homework 9 is due Tuesday, November 11, at the beginning of lecture. The usual considerations apply.

In 175 we will try to cover this week until section 8.4 at least, but probably we won’t get there until next week. The key section here is 8.2; make sure you understand the notions discussed in 8.2 before going further. If you want to read ahead from 8.4, continue with sections 8.5 and 8.6; the difference between conditional and absolute convergence is very important here.

In 275 we will cover from section 13.4 on, and the goal is to reach 13.8, which probably won’t happen until next week or even the one after if things do not go well. Besides these topics, I will discuss the `mean value property’ of harmonic functions.

Homework 9:

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

Section 8.1. Exercises 86, 88, 127. Also, the following exercise:

Starting with a given , define the subsequent terms of a sequence by setting . Determine whether the sequence converges, and if it does, find its limit. More precisely: You must indicate for which values of the sequence diverges, and for which it converges, and for those that converges, you must identify the limit, which may again depend on . You may want to try studying the sequence with different initial values of (choose a large range of possible values) to get a feeling for what is going on.

Section 8.2. Exercises 14, 22, 38, 40 (do not use a calculator for this one; you can use that if necessary), 64-68, 71.

Section 8.3. Exercises 26, 35, 41, 43, 44.

There are 19 problems in total. Turn in at least 10. The others (at most 9) will be due November 18 together with a few additional exercises for that week. I suggest you start working on these problems early, as some may be a bit longer than usual.

I thought about this question a while ago, while teaching a topics course. Since one can easily check that $${}|{\mathbb R}|=|{\mathcal P}({\mathbb N})|$$ by a direct construction that does not involve diagonalization, the question can be restated as: Is there a proof of Cantor's theorem that ${}|X|

First of all, note (as Monroe does in his question) that if $\mathbb P,\mathbb Q$ are ccc, then $\mathbb P\times\mathbb Q$ is $\mathfrak c^+$-cc, as an immediate consequence of the Erdős-Rado theorem $(2^{\aleph_0})^+\to(\aleph_1)^2_2$. (This is to say, if $\mathbb P$ and $\mathbb Q$ do not admit uncountable antichains, then any antichain in their product ha […]

The technique of almost disjoint forcing was introduced in MR0289291 (44 #6482). Jensen, R. B.; Solovay, R. M. Some applications of almost disjoint sets. In Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), pp. 84–104, North-Holland, Amsterdam, 1970. Fix an almost disjoint family $X=(x_\alpha:\alpha

At the moment most of those decisions come from me, at least for computer science papers (those with a 68 class as primary). The practice of having proceedings and final versions of papers is not exclusive to computer science, but this is where it is most common. I've found more often than not that the journal version is significantly different from the […]

The answer is no in general. For instance, by what is essentially an argument of Sierpiński, if $(X,\Sigma,\nu)$ is a $\sigma$-finite continuous measure space, then no non-null subset of $X$ admits a $\nu\times\nu$-measurable well-ordering. The proof is almost verbatim the one here. It is consistent (assuming large cardinals) that there is an extension of Le […]

R. Solovay proved that the provably $\mathbf\Delta^1_2$ sets are Lebesgue measurable (and have the property of Baire). A set $A$ is provably $\mathbf\Delta^1_2$ iff there is a real $a$, a $\Sigma^1_2$ formula $\phi(x,y)$ and a $\Pi^1_2$ formula $\psi(x,y)$ such that $$A=\{t\mid \phi(t,a)\}=\{t\mid\psi(t,a)\},$$ and $\mathsf{ZFC}$ proves that $\phi$ and $\psi […]

A notion now considered standard of primitive recursive set function is introduced in MR0281602 (43 #7317). Jensen, Ronald B.; Karp, Carol. Primitive recursive set functions. In 1971 Axiomatic Set Thoory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) pp. 143–176 Amer. Math. Soc., Providence, R.I. The concept is use […]

The power of a set is its cardinality. (As opposed to its power set, which is something else.) As you noticed in the comments, Kurepa trees are supposed to have countable levels, although just saying that a tree has size and height $\omega_1$ is not enough to conclude this, so the definition you quoted is incomplete as stated. Usually the convention is that […]

The key problem in the absence of the axiom of replacement is that there may be well-ordered sets $S$ that are too large in the sense that they are longer than any ordinal. In that case, the collection of ordinals isomorphic to an initial segment of $S$ would be the class of all ordinals, which is not a set. For example, with $\omega$ denoting as usual the f […]