Nonstandard Analysis (1972)

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 you how simple first-year analysis is when approached this way (but it doesn't go much into what it actually is).

(Of course, per Thomas Forster IIRC, the purpose of a first-year course in real analysis is not actually to teach you about real analysis; so in my view this rather defeats the point. First-year real analysis is taught primarily to make sure you understand the difference between "for all x there exists y" and "there exists x such that for all y".)

IMO this is a good book on nonstandard analysis: https://eml.berkeley.edu/~anderson/Book.pdf

Infinitesimal Methods in Mathematical Economics by Robert Anderson, math and econ professor at Berkeley

Terence Tao's blog entry "Nonstandard analysis as a completion of standard analysis"

https://terrytao.wordpress.com/2010/11/27/nonstandard-analys...

I recognize this content from Hersh and Davis's later book "The Mathematical Experience," which I really like.

Mirror: https://gwern.net/doc/math/1972-davis-2.pdf