Regarding your parenthetical final paragraph: As the surreals are totally ordered, and polynomials only have finitely many roots, we can canonically choose the smallest root of a polynomial (or the largest).

My last comment -- Silver, Solovay and Kunen (see section 3.6 of Solovay's article Real-Valued Measurable Cardinals from the book Axiomatic Set Theory) proved that measurable cardinals, if consistent, are compatible with the generalized continuum hypothesis, and therefore the continuum hypothesis. So the answer to question 2 is no, because measurable cardinals are real-valued measurable. However, if the continuum hypothesis holds and there is a metric space $X$, with a non-separable probability measure on it, $X$ must have cardinality very much larger than the continuum.

One more thing -- the first measurable cardinal is weakly inaccessible, so the continuum is measure-free not only if $2^{\aleph_0} = \aleph_1$, but also if it is $\aleph_2$, $\aleph_3$, $\aleph_{\epsilon_0 + 1}$, $\aleph_{\omega_5}$, etc.

Billingsley also undersells the following point (as do all analysis books of that era when measurable cardinals come up, such as Gillman & Jerison's Rings of Continuous Functions and Schaefer's Topological Vector Spaces) -- it is not really an "unsolved problem" as to whether real-valued measurable cardinals exist, rather the existence of real-valued measurable cardinals proves the consistency of ZFC, so they cannot be proven to exist, nor even to be relatively consistent, if we just start with ZFC. A nice textbook reference for these facts is Jech's "Set Theory: Third Millennium Edition".

Additionally, the terminology Billingsley uses for measurable cardinals, although it is a direct translation of Ulam's terminology in the original paper, is out of date. Nowadays we say that a cardinal $\kappa$ is real-valued measurable if there is a $\kappa$-additive probability measure $\mu$ on $\kappa$, considered as a discrete metric space, such that $\mu(\{x\}) = 0$ for all $x \in \kappa$. It is then a theorem that the smallest cardinal with a countably additive probability measure vanishing at every point is the first real-valued measurable cardinal.

Appendix III, the reference to Keisler and Tarski, and Theorem 2 do appear in my copy of the 1968 original from Wiley and Sons. Which English original were you looking at?

@JoelDavidHamkins I've just checked -- Hodges uses it in his book Building Models by Games, at the bottom of page 23. This was published in 1985.

The use of Abélard and Éloïse is due to Wilfred Hodges. See page 6: wilfridhodges.co.uk/semantics06.pdf His undergraduate degree was in theology.

In Schaefer's book, Topological Vector Spaces, sections II.5-II.8 deal with limits and colimits in locally convex spaces and facts such as every complete locally convex space being a directed limit of Banach spaces, and every bornological space being a directed colimit of normed spaces. The terminology is old-fashioned, however, and category theory is not used directly.

@gondolf As Nik says, this is not research level, so you can find the answer in a textbook. For example, Appendix II of Dixmier's Von Neumann Algebras.

@gondolf That doesn't affect Nate's counterexample. If $A$ is a trace-class operator, so is $nA$ for any $n \in \mathbb{N}$, and its trace is $n$ times the trace of $A$. You need to put an upper bound on the sequence.

Although Weil based $\emptyset$ on ø because he encountered it in Norwegian, the letter ø did not originate in Norwegian. In fact, Norwegian did not start becoming a written language until the nineteenth century (if Norwegians needed to write something before then, they would use Danish). Exactly which language ø originated in seems to be lost to history, however.

@TomEllis Cipher is also an old-fashioned word for zero in English, and is the most similar to the Arabic sifr. Since Grothendieck was talking about the introduction of zero, and how it would have seemed at the time, he may have used the old name for it on purpose for rhetorical effect (this is how I interpreted it). Incidentally, his Muttersprache was German.

@bof I should emphasize that I do not take that interpretation of the word "separable" myself, or in my answer, and I am well aware that second countability is hereditary for subspaces. Maybe your comment should be directed to Fedor Petrov?