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I think there are many examples. The Rado graph plus an isolated point, or plus $n$ isolated points, the disjoint union of two Rado graphs, the disjoint union of all finite graphs . . .
Nice question. For a differentiable function $f$ on $[0,1]$ (or even just Lipschitz), there will be some $m$ large enough that $g(x) = f(x)+mx$ and $h(x) = f(x)-mx$ are both injective. By choosing larger and larger $m$'s on the intervals $[n,n+1]$, you can probably get every differentiable function $\mathbb R \rightarrow \mathbb R$ as a sum of two Borel injections. The same idea doesn't quite work for general continuous functions, though.
"More likely, X's beliefs would agree with fictionalism in some ways and would disagree in other ways." I couldn't agree more. I've read a little bit about fictionalism, and some aspects of it resonate strongly with me. I might even be more of a fictionalist than a platonist. Nevertheless, I would not label myself as a fictionalist -- it seems too much like signing a contract I've not read all the way through.
@GabeGoldberg: I wouldn't count that one as obvious to me. I can't say whether it would be obvious to someone who's less ignorant of set-theoretic geology.
I suppose any principle implying the Ground Axiom is destroyed by every set-sized forcing. Already I think it's an interesting question to ask what set-theoretic principles, other than the obvious ones, are always destroyed by set-sized forcing. (The "obvious" ones, to me, are things like $V=L$ or $V=L[\mu]$.)
Maybe I misunderstood your definition -- do you allow $n=0$ in your definition of $C$? If so (as I assumed) then the map you're describing has $2^{\aleph_0}$ equivalence classes. If not, I think you want to change your description of the poset you're talking about, so that you're not looking at a relation on the whole diagonal of $X$, but only the points $x \in X$ such that $x C x$.
Hi Mike, welcome to MO. If you'd like to clarify your question further, or include some more of what you know about it, then the normal way to do that is to edit the question and include that information as part of the question. I hope you don't mind -- I'm going to edit your question and move the data from the answer you posted so as to make it part of the question.
