Comment by ajkjk

I guess my stance (which is not very well-developed or anything) is that you try to learn to live with the gaps: define everything in terms of only what you can measure and it no longer matters whether a number is rational or irrational, or infinitesimal vs small-but-finite, because you can't tell. Instead of saying "it vanishes" as an absolute statement you say "it appears to vanish from my perspective".