Non-measurable sets are precisely the kinds of things Arnold wanted marginalized in mathematical pedagogy, instead of placed front and center. They are necessary auxiliaries to the main theory, that of measures and integration but auxiliary nonetheless.

I disagree that transfinite numbers are detached from physical intuition since most of the ones you or I could write down can be easily visualized with a few ellipses here or there. But i do think Arnold would consider them marginal players. Perhaps he thought set theory was a formalist distraction from the main of mathematics!