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> If you want to go beyond calculus, almost everything (in this and related areas) is about sequences, limits, limiting processes, functions, and transformations. There, non-standard analysis tends not to help ... Why do you say this? I ask because I've found internal set theory, Edward Nelson's axiomatic version of nonstandard analysis, to be a lovely tool for doing typical sorts of things in analysis. You have to learn to wield the "standard" predicate [0], which is too dark an art for some mathematicians, I suppose. But, in my opinion, nonstandard characterizations of notions like convergence and continuity are delightfully simple and direct. It also turns out that when you have nonstandard numbers at hand, infinity is an over-powerful abstraction for some purposes. Nelson came up with a new formalism for probability theory [1], for example, that makes finite spaces powerful enough to capture what's interesting for most purposes. Similarly, finite but unlimited sequences often are "long enough" to incorporate all the interesting behavior of infinite sequences. 0. Alain Robert's Nonstandard Analysis is a good starting point. 1. See his short book Radically Elementary Probability Theory. I love this book, and didn't much like probability theory before reading it. |