超实数讲义
Lectures on the hyperreals
Robert Goldblatt · 初版 1998 年
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本书是关于 nonstandard analysis 的入门介绍，基于作者多次讲授的讲义整理而成。它适合作为研究生低年级或本科高年级的教材，也适合任何熟悉初等实分析的人自学。它不仅将 nonstandard analysis 作为关于无穷小和无穷大数的理论来呈现，更将其作为审视诸多标准数学概念与构造的截然不同的全新视角；它是新思想、新对象和新证明的源泉；也是强大推理新原则（transfer、overflow、saturation、en
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……我觉得）Hurd 和 Loeb 的《An Introduction to Nonstandard Real Analysis》里讲了一些积分。至于非标准分析的基础以及广泛的综述，我强烈推荐 Robert Goldblatt 的《Lectures on the Hyperreals: an Introduction to Nonstandard Analysis》，这是 GTM（Graduate Texts in Mathematics）系列里的一本。
…think) some integration, in Hurd and Loeb, "An Introduction to Nonstandard Real Analysis". For the foundations of nonstandard analysis, and a wide-ranging overview, I strongly recommend Robert Goldblatt, "Lectures on the Hyperreals: an Introduction to Nonstandard Analysis", one of the Graduate Texts in Mathematics.
关于非标准分析，我读过最好的书是 Goldblatt 写的《Lectures on the Hyperreals》，写得绝了。从更偏实用的角度来说，Pétry 的《Analyse Infinitésimale: une présentation non standard》是一本极具可读性的大一本科生水平教材，它展示了……
The best book I have read on nonstandard analysis is "Lectures on the Hyperreals", by Goldblatt, which is excellent. On the more practical end, Pétry's "Analyse Infinitésimale: une présentation non standard" is an extremely readable first-year-undergrad level textbook that shows…
顺便说一句，涉及非标准实数的计算是正确的（凭我十年前学这玩意儿的记忆来看）。老规矩，如果要了解这一切是如何运作的入门书，我推荐 Goldblatt 的《Lectures on the Hyperreals》；如果要看用非标准分析讲的本科初阶分析课程，我推荐 Pétry 的《Analyse Infinitésimale: une présentation non standard》。
By the way, the computation involving nonstandard reals is correct (to the best of my ten-year-old memory of studying this stuff). As usual, I will recommend Goldblatt's _Lectures on the Hyperreals_ for an intro to how it all works, and Pétry's "Analyse Infinitésimale: une présentation non standard" for an undergraduate first course in analysis expressed through nonstandard analysis.
如果我们走超实数路线的话，我相当喜欢 Goldblatt 那本 GTM 的《Lectures on the Hyperreals》。如果你想处理其他非标准对象，得再补充一两篇论文，但我读研那会儿，每次想理清思路都会反复回去翻它。
If we're going the hyperreal route, I quite like Goldblatt's GTM Lectures on the Hyperreals. You have to augment it with a paper or two if you want to work with other nonstandard objects, but when I was doing my graduate work it was the resource I kept going back to for clarity.
……the History of Topos Theory》[2]（这本不需要读过 [1]，它只是推翻了 [1] 中的一些历史假设，而且和 McLarty 的其他作品一样，写得极其出色）3. Goldblatt 的《Lectures on the Hyperreals》[3] 4. Nelson 的《Radically Elementary Probability Theory》[4] 5. 陶哲轩的《Ultraproducts as a Bridge Between Discrete and Continuous Analysis》[5] 6. 某本经典的机器学习教材，比如 Murphy 的那本[6]……
…the History of Topos Theory"[2] (this does not require [1], it just undoes some historical assumptions made in [1] and, like everything else by McLarty, is extraordinarily well-written) 3. Goldblatt's "Lectures on the Hyperreals"[3] 4. Nelson's "Radically Elementary Probability Theory"[4] 5. Tao's "Ultraproducts as a Bridge Between Discrete and Continuous Analysis"[5] 6. Some canonical machine learning text, like Murphy[6]…