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calef comments on Approaching Infinity - Less Wrong
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Fair enough, I was simply trying to appeal to what is probably his most familiar intuition regarding the real number line.
Most people that are going to have confusion about a big number being close to infinity probably aren't going to know what a metric space is.
They may not know the term "metric space" (in which case you just explain that it's a setting where we can measure distance), but if they think larger numbers are "closer to infinity" than smaller numbers, that means they are intuitively thinking in terms of the extended real line (metrized in one of the usual ways).