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John_Maxwell_IV comments on New Year's Predictions Thread - Less Wrong
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Actually, I'm fine with people speaking vaguely, I just don't want to see terminology misused.
"Through adding zeroes between the decimal point and the 7 in the string '.7', the number we are representing can be made arbitrarily small." Is this a misuse of the word "arbitrarily"?
The important think about an epsilon in a mathematical proof is, conventionally, that it can be made arbitrarily small. This is a human interpretation I am adding on to the proof itself. If the important thing about a variable in a proof was that the variable could become arbitrarily large, my guess is that a variable other than epsilon would not be used.
Your usage is fine, so long as it's clear that "arbitrarily small" is a feature of the set from which you are choosing numbers, or of the process by which you are constructing numbers, and not of any particular number in that set. This is clear with the context that you give above. It wasn't as clear to me when you wrote that "epsilon is (conventionally) an arbitrarily small number".