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Asaf, "most sets are infinite" is ambiguous precisely because we do not yet agree on a way to measure sub-classes of the class of all sets (independently of whether or not we agree to treat two sets as equal in this class if there exists a set-bijection between them). The question asks whether there is a measure theory on classes which provides some framework for making such statements non-ambiguous.
Asaf, either you don't get my question or I don't get your comments. Saying that measure theoretically there are as many integers as Cantor points assumes the existence of a common measure space (maybe Reals with Lebesgue measure?) into which both parties can inject, and then you use that implicitly assumed measure to compare. The fact that another measure (or another common space) might yield a different answer hardly undermines the usefulness of measure theory itself.
Zsbán, a delay differential equation would be something like $f'(x) = f(f(x-a))$. The right side of the equation above does not exhibit dependence on the backward/forward trajectory and hence there is no non-zero "delay"
Tom, thank you very much for this answer. But how does it follow that $X$ is a compact $n$-manifold with boundary in $n$-space? For example, the minimal CW decomposition of the $2$-sphere embeds in $\mathbb{R}^3$ but surely not as a compact $3$-manifold, so I must have misunderstood something in the "without loss of generality" part.
