Intro to IST and Nonstandard Analysis

2011 July 5
by Daniel Lakeland

Dover has just published their edition of Alain Robert's book on Nonstandard Analysis (NSA). This book is based on Internal Set Theory, a simplified axiomatic version of NSA. Those who follow the blog may remember that one of my interests is in using Nonstandard Analysis to make derivations of physical models easier and more precise, especially physical models with multiple scales.

Continuous models of real phenomena are almost always wrong, for example the world is made up of discrete atoms. So interpreting a continuous model as taking a limit as a small spatial quantity dx goes to zero can't be right. For example the density of matter in a fluid has a limit as dx goes to zero of zero density, since if we get our box small enough there will be no atoms or even electrons in it. However there are ways around this if you change your interpretation, and NSA makes these new interpretations fairly reasonable.

Those with an interest in multiscale models, or interpretations of continuum models of discrete systems should check out this book.

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