This set is due April 3 at the beginning of lecture. Details of the homework policy can be found on the syllabus and here.

1. Find where and determine all its subfields. Make sure you justify your answer. For example, if you state that two subfields and are different, you need to prove that this is indeed the case.

2. Do the same for

[Updated, April 2: I guess the hint I gave for problem 2 makes no sense, sorry about that. Rather, you may want to begin by looking at how factors. Then, to compute it may be helpful to look at a triangle with angles and ]

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[…] Homework 6, due April 3, at the beginning of lecture. Possibly related posts: (automatically generated)117b – Homework 5Buffalostyle Forges OnHomework battles and the biggest genius in the school, part IThe Myth About Homework […]

Concerning HW assignment 6, I am wondering if there is a typo in the hint you gave us for problem number 2. I may be wrong but I believe the denominator under the radical on the left hand side should be 18. Thanks!

Hi,
Okay…I am struggling with the second problem. I have solved the quartic down to w^6 and found my value for w. Then, when I plug everything back in, I cannot get any given u to solve the equation where u^3 + ….=0. Therefore I am second guessing everything I have done. And I appreciate the new hint, but I’m not sure how to apply it. I have worked the equation many times, each time hoping my calculations are wrong…unfortunately so far, they are not.
Any help would be greatly appreciated!

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 […]

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 […]

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 $\psi$ […]

[…] Homework 6, due April 3, at the beginning of lecture. Possibly related posts: (automatically generated)117b – Homework 5Buffalostyle Forges OnHomework battles and the biggest genius in the school, part IThe Myth About Homework […]

Concerning HW assignment 6, I am wondering if there is a typo in the hint you gave us for problem number 2. I may be wrong but I believe the denominator under the radical on the left hand side should be 18. Thanks!

Hi Tommy,

Hmm, yeah. I’m just about convinced now that the hint is nonsense. So, I have added another hint.

Hi,

Okay…I am struggling with the second problem. I have solved the quartic down to w^6 and found my value for w. Then, when I plug everything back in, I cannot get any given u to solve the equation where u^3 + ….=0. Therefore I am second guessing everything I have done. And I appreciate the new hint, but I’m not sure how to apply it. I have worked the equation many times, each time hoping my calculations are wrong…unfortunately so far, they are not.

Any help would be greatly appreciated!