April 9, 2011
I think I’m done talking about NSA for now. But here’s all of it in one place, and a couple links for further reference.
1. “Free Ultrafilters” cigarette ad.
2. Slide show, intended to also be self-explanatory survey (for people who’ve been through “Baby Rudin”) For a while, NSA looked kind of redundant and pointless, but the proof of Tychonoff’s theorem was nice and short. It seems like things that depend heavily on the Axiom of Choice might be good places to apply NSA. Whether it’s a good idea to have a whole functional analysis course using NSA in places might be questionable. The slide show itself came out to only a half hour (going fast), so if I were to lengthen it, I might have added a proof of Robinson’s Theorem. The Transfer Principle might have been too much, but if I understood it better, I would at least include some big-picture ideas of why it’s true. A lot of *-transferred sentences are pretty obviously true, though, like the nonstandard Archimedean property.
4. Remark – NSA isn’t that fashionable, and large portions of it seem sort of almost circular – you take the sequence arguments you already know and love, encapsulate it in some special notation with equivalence classes, and write up the same proof with different notations. The most important inequality will probably have to be transferred back and forth at some point, but it will certainly feel very familiar but scrambled a bit. It could be that either you need to get deeper into it to get any real payoff, or it could be that Axiom of Choice management in some settings is simplified by embedding the AC in the definitions of your basic objects.
April 4, 2011
These slides are based on A.E. Hurd, P.A. Loeb – An Introduction to Nonstandard Analysis.
Check them out. They’re supposed to be moderately self-contained, but standard analysis basic fluency is somewhat assumed.
EDIT: New draft. Some more examples, figures.
EDIT2: Presented this at the student analysis seminar. Found some typos, naturally. Fixed for posterity and for great justice.