So, we went to Kathryn Albertson Park, to play football. But it was so hot, and there were so many geese mementos on the grass, that Francisco felt unhappy, and wanted to go home instead.

We are walking back to the car, when

“Wait. Is that a Dalek?” Francisco was ahead of me. He stopped, came back, and looked at where the camera and I were pointing.

“Yes. That’s a Dalek. And that’s the Doctor.”

The part that I wasn’t expecting was what happened next. He looked at Najuma and I, scared, and said:

“I want to get in the car. Let’s go home.” He started to go for the car, then he looked at me. Why are you not running, you fool?

“No, wait, let me take another picture.”

“No, no, let’s go. Let’s go…” There is a Dalek in the park coming for us, you people, what are you doing? Run! RUN!!

“Oh, it’s talking, what is it saying?”

What else, really? EX-TER-…

So, yeah. We got in the car and sped out of there and into safety.

“Is the Dalek following us, papi?”

“No, I don’t think it is.”

“Are we safe at home?”

“Sure we are. And anyway, let me tell you, I’ll protect you of any Dalek attacks we may suffer, ok?”

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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.

As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]

Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0

Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]

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 […]

The question seems to be: Assume that there is a Vitali set $V$. Is there an explicit bijection between $V$ and $\mathbb R$? The answer is yes, by an application of the Cantor-Schröder-Bernstein theorem: there is an explicit injection from $\mathbb R$ into $\mathbb R/\mathbb Q$ (provably in ZF, this requires some thought, or see the answers to this question) […]

If a set $X$ is well-founded (essentially, if it contains no infinite $\in$-descending chains), then indeed $\emptyset$ belongs to its transitive closure, that is, either $X=\emptyset$ or $\emptyset\in\bigcup X$ or $\emptyset\in\bigcup\bigcup X$ or... However, this does not mean that there is some $n$ such that the result of iterating the union operation $n$ […]

I PLEDGED NOT TO EXTERMINATE TODAY!

Oh, how fantastic! And through Dalek Klaus’s twitter account, I found https://www.facebook.com/media/set/?set=a.10151472718672638.1073741852.95835192637&type=3&l=6282565f58 (148 pictures of the adventure).

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