We verified that the sets form a continuous increasing sequence and are transitive. It follows that the reflection theorem holds for the and . Arguing in , we proved that is a model of , and the reflection theorem allowed us to simplify the proof in a few points.

We then proceeded to argue that is also a model of choice. In fact, there is a globally definable well-ordering of . It is worth emphasizing that the well-ordering is a very natural one, as we simply proceed to enumerate the sets in in the order in which their membership is verified. The definitions of the sequence of sets and of this well-ordering are absolute, and we used this to prove that is a model of the statement “,” and so is any , for limit. Moreover, the well-ordering of , when restricted to , coincides with its interpretation inside .

An easy induction shows that for infinite, . An argument using the Mostowski collapsing theorem allowed us to prove Gödel’s condensation lemma: If for a limit ordinal, then is isomorphic to some . These two facts combine to provide a proof that holds in .

Remark. These arguments prove that implies , but they also indicate that showing that implies ought to be more complicated. The reason is that the absoluteness of the construction of implies that if is a transitive proper class model of , then and in fact , i.e., the result of running the construction of from the point of view of is itself. But, since holds in , we cannot prove in that there is a non-constructible set. If we tried to establish the consistency of with the negation of choice by a similar method, namely, the construction of a transitive class model of , then running the construction inside would give us that , so , which would be a contradiction, since we are assuming that (provably in ) is a model of but is a model of choice.

This also suggests that in order to show that is independent of , one should try first to show that is consistent with . The remarkable solution found by Paul Cohen in 1963, the method of forcing, allows us to prove the consistency of both statements, and also to do this while working with transitive models. The method of forcing is beyond the scope of this course, but good explanations can be found in a few places, there is for example a book by Cohen himself, or look at Kunen’s book mentioned at the beginning of the course. Richard Zach has compiled in his blog a list of papers providing an introduction to the method (search for `forcing’).

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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 […]

You assume $\omega_\alpha\subseteq M$ and $X\in M$ so that $X$ belongs to the transitive collapse of $M$ (because if $\pi$ is the collapsing map, $\pi(X)=\pi[X]=X$. You assume $|M|=\aleph_\alpha$ so that the transitive collapse of $M$ has size $\aleph_\alpha$. Since you also have that this transitive collapse is of the form $L_\beta$ for some $\beta$, it fol […]

No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]

I don't think you need too much in terms of prerequisites. An excellent reference is MR3616119. Tomkowicz, Grzegorz(PL-CEG2); Wagon, Stan(1-MACA-NDM). The Banach-Tarski paradox. Second edition. With a foreword by Jan Mycielski. Encyclopedia of Mathematics and its Applications, 163. Cambridge University Press, New York, 2016. xviii+348 pp. ISBN: 978-1-10 […]

For the second problem, write $x=-3+x'$ and so on. You have $x'+y'+z'=17$ and $x',\dots$ are nonnegative, a case you know how to solve. You can also solve the first problem this way; now you would set $x=1+x'$, etc.