I ran into this picture a few days ago, when looking at the old photos from the Martin Conference. I posted here the group picture from that conference. John Steel should be posting the other pictures soon (well, I’ve been waiting since 2001, so we’ll see), or I may post them if that ends up not happening.

This one is the conference photo from the Logic Colloquium 2000. The website has many other pictures available as well. It was an interesting meeting.

Paris, July 23 – 31, 2000. The meeting site will be the Sorbonne, where David Hilbert presented his famous list of problems at the International Congress of Mathematicians in August 1900.

As I recall, at the opening we listened to a recording of part of Hilbert’s address, including his Wir müssen wissen. Wir werden wissen.

I like this picture very much. You can see me behind Joel Hamkins, in what seems to be row eight. (I believe I met Joel and his wife, Barbara Gail Montero, at this conference.) Paul Larson still has hair.

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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 […]

Paul, via email: “By the way, I still have hair. Just not everywhere.”