(This post is specifically for Math 287 students.)

Starting Monday, you guys should be organized in new groups. No group can have three members that are together in the current groups. When I arrive on Monday, the groups should already be formed. You guys should start working on the laboratory on Polyhedra, Chapter 7. Make sure to bring whatever materials may be needed for this.

There have been complaints about not everybody contributing their share to their respective groups. This is not acceptable, but it is sadly the main reason for the reorganization. So: If I receive two complaints about a member not contributing as required, and there are no reasonable extenuating circumstances, that person will be dropped out of their group (receiving a zero in their current project as a result). If the issue is not expected to be resolved for the next report, a new reorganization of groups will be triggered as a result.

Also, I am unhappy with the level of some of the reports. It seems the peer reviewing of other groups’ drafts is not being taken as seriously as needed. So: Now, on the dates drafts are due, each group should bring three copies of their current draft. When you review another group’s draft, write the members of your group on the copy you are reviewing. I’ll collect the copies, with your comments, copy them for my records and return. If I identify something that a group should have noticed and mentioned, but did not, that group will be penalized (since this means the group did not take their refereeing role seriously). Conversely, if a group mentions something that should be addressed, but I do not see the issue resolved in the final report, the group that failed to address the given comments will be penalized.

Finally, this being a mathematics course, I expect your projects to include proofs. If a project lacks proofs it will receive a failing grade.

Feel free to contact me be email if any of the above needs clarifying.

[…] de puissances, Fundamenta Mathematicae, 3, (1922), 52–58, or Fritz Herzog, George Piranian, Sets of convergence of Taylor series. I. Duke Math. J., 16, (1949), 529–534. Both papers prove more general results, by explicit […]

Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter does not imply the existence of a Vitali set. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${ […]

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 […]

Equality is part of the background (first-order) logic, so it is included, but there is no need to mention it. The situation is the same in many other theories. If you want to work in a language without equality, on the other hand, then this is mentioned explicitly. It is true that from extensionality (and logical axioms), one can prove that two sets are equ […]

$L$ has such a nice canonical structure that one can use it to define a global well-ordering. That is, there is a formula $\phi(u,v)$ that (provably in $\mathsf{ZF}$) well-orders all of $L$, so that its restriction to any specific set $A$ in $L$ is a set well-ordering of $A$. The well-ordering $\varphi$ you are asking about can be obtained as the restriction […]

Gödel sentences are by construction $\Pi^0_1$ statements, that is, they have the form "for all $n$ ...", where ... is a recursive statement (think "a statement that a computer can decide"). For instance, the typical Gödel sentence for a system $T$ coming from the second incompleteness theorem says that "for all $n$ that code a proof […]

When I first saw the question, I remembered there was a proof on MO using Ramsey theory, but couldn't remember how the argument went, so I came up with the following, that I first posted as a comment: A cute proof using Schur's theorem: Fix $a$ in your semigroup $S$, and color $n$ and $m$ with the same color whenever $a^n=a^m$. By Schur's theo […]

It depends on what you are doing. I assume by lower level you really mean high level, or general, or 2-digit class. In that case, 54 is general topology, 26 is real functions, 03 is mathematical logic and foundations. "Point-set topology" most likely refers to the stuff in 54, or to the theory of Baire functions, as in 26A21, or to descriptive set […]

[…] de puissances, Fundamenta Mathematicae, 3, (1922), 52–58, or Fritz Herzog, George Piranian, Sets of convergence of Taylor series. I. Duke Math. J., 16, (1949), 529–534. Both papers prove more general results, by explicit […]