People sometimes use the term "finite" to mean "non-zero" or "non-infinitesimal". For example, physicists often say "finite temperature" to emphasize that the temperature under consideration is not zero. Similarly, when speaking of symmetry groups one might say "finite translation" to emphasize that a translation is not infinitesimal. This is not strictly a correct usage of the term finite, which technically encompasses zero and infinitesimal values just as well as the rest. So my question is: Is there a better word to use in these cases, to refer to non-zero, non-infinitesimal, but still finite values?
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$\begingroup$ How about a non-zero constant? $\endgroup$– StudentmathCommented Dec 6, 2014 at 23:34
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1$\begingroup$ A nonzero real number? By the fact that it is a real number it is finite, and by the fact that it is nonzero, there are many numbers smaller than it (in magnitude). $\endgroup$– JMoravitzCommented Dec 6, 2014 at 23:34
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$\begingroup$ The difference between "finite transformations" and "infinitesimal transformations" has little to do with numbers. Physicists refer to an element of the Lie algebra of a Lie group as an infinitesimal transformation. They aren't actually elements of the group, though they do map to elements of the group via the exponential map. $\endgroup$– Matt SamuelCommented Dec 6, 2014 at 23:37
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1$\begingroup$ "Finite nonzero" works for me. $\endgroup$– Qiaochu YuanCommented Dec 7, 2014 at 2:02
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1$\begingroup$ @Sesquipedal: Goldblatt (Lectures on the Hyperreals, p. 50) calls such numbers "appreciable" (= "limited but not infinitesimal"). $\endgroup$– Hans LundmarkCommented Dec 7, 2014 at 15:09
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On p. 50 in Goldblatt's book Lectures on the Hyperreals, hyperreal numbers $b$ which are "limited but not infinitesimal" ($r<|b|<s$ for some $r,s \in \mathbb{R}^+$) are called appreciable.
answered Dec 8, 2014 at 7:10