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@EmilJeřábek: Good point. What if we go the other direction and use the upward L-S theorem to get a compact metric space of size $>\mathfrak{c}$? This is also impossible -- every compact metric space has size $\leq\!\aleph_0$ or $\mathfrak{c}$.
If we could define "compact" in the language you've described, then we could also define "compact, nonempty, and has no isolated points" (since the latter two properties are easy to describe in this language). But then the downward Lowenheim-Skolem theorem would give us a countable metric space with these propoerties. This is a contradiction, because every compact, nonempty metric space with no isolated points has size $\mathfrak{c}$. (I'm not sure I understand your question, which is why I'm making this a comment and not an answer. Does this answer your question?)
@ChristianRemling: That is a valid point. But I think that if $f$ is continuous when restricted to a full-measure set, then there is an a.e. continuous function $g$ that is a.e. equal to $f$. (See mathoverflow.net/questions/145957/…) Combined with this post, I think that answers the original question.
OK, such permutations do exist. To get one, you can take a partition of $\mathbb N$ into longer and longer intervals, with the lengths of the intervals growing very fast. Then find a permutation $\sigma$ that never maps any number outside of the interval it starts in, but does jumble up each of the intervals internally. Such a permutation could make a conditionally convergent series diverge by oscillation, but could never make it converge to a different value (because the finite partial sums of the rearrangement equal those of the original series infinitely often, at the end of each interval).
In your last paragraph, you talk about "permutations which send convergent series to non convergent series, but which don't send any convergent series to a convergent series with a different value". Are you certain that such permutations really exist?
Let me add that for certain Borel forcings, the Martin number corresponds to a well known cardinal characteristic of the continuum. For example, the Martin number for the Cohen forcing is the covering number of the meager ideal. Thus, if $\mathbb P$ is the Cohen forcing then $\text{FA}(\kappa,\mathbb P)$ holds if and only if $\mathrm{cov}(\mathcal M) > \kappa$. Similarly with the random forcing and $\mathrm{cov}(\mathcal L)$, or with the Laver forcing and $\mathfrak{b}$, or with the Miller forcing and $\mathfrak{d}$, etc.
In my mind, one of the most important aspects of compactness is that a compact topology (in an abstract "pointless" sense) can be viewed as having been built from finite collections of open sets. For example, compact zero-dimensional Hausdorff spaces are precisely those that are inverse limits of finite discrete spaces. I don't know much category theory, but this feels to me like something that might generalize to other categories.
@JukkaKohonen: An even easier example to show this, if you don't require the graph to be connected (an Alaska/Hawaii sort of situation) is just to take the disjoint union of two maps that each admit a "good" numbering.
I think your question is equivalent to asking whether a certain graph has a Hamiltonian path, the graph formed by taking each region of the map as a vertex with two regions connected if and only if they are adjacent. So you might learn something of interest to you by reading about Hamiltonian paths. en.wikipedia.org/wiki/Hamiltonian_path
I'm having trouble understanding what the word "epistemic" is doing in the title of this question. It seems to make perfect sense with that word deleted, and less sense (to me) with it there. Do you mind clarifying?
@MatteoCasarosa: Yes, I usually think of the $\leq^*$ version of Hausdorff gaps as being a bit stronger, since the other version is easily obtained from it. I don't know how to go the other way, but I also don't know of any theorem that suggests it's impossible to.
@MatteoCasarosa: Yes, Hausdorff's construction gives you that these sets are always infinite. I guess that the definition I wrote down doesn't prescribe this, but you could alter the definition, if you like, by replacing $\subseteq^*$ with $\subset^*$.
