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ultrasmall vs infinitesimal

Why "ultrasmall" rather than "infinitesimal"?

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Historically, mathematicians have used the word "infinitesimal" for a long time. In the informal understanding of the fathers of calculus (Newton, Leibniz, Bernoulli, Euler, etc.,) this conveyed the idea of vanishingly small quantities.

In modern mathematics, an infinite set is a set that can be put in one-to-one-correspondence with a proper subset of itself.  In the 1960's, Robinson constructed the semiring of hypernatural numbers, an extension of the familiar semiring of natural numbers. In this extension there exist numbers greater than every natural number. These nonstandard numbers are infinite in the usual sense (more precisely, the set of all smaller hypernaturals is infinite), and therefore it makes sense to call them "infinitely large natural numbers."

In our view of analysis with ultrasmall numbers, the familiar set of natural numbers has additional ideal, ultralarge elements that are inseparable from the usual, ''real'' ones (for a discussion of this point, see here). These ultralarge natural numbers have the flavor of Robinson’s infinitely large numbers; the difference is that we regard them as natural numbers (albeit ideal), with all the usual properties of natural numbers.  In particular, the set of all smaller natural numbers is finite (although of ultralarge size). Using the word "infinite” for these numbers would contradict the usual meaning of the word and facilitate erroneous conclusions. For compatibility with this terminology, we call the reciprocals of ultralarge numbers ultrasmall rather than ''infinitesimal.'' We also say that two numbers are ultraclose rather than ''infinitesimally close.''