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Showing posts from August, 2022

Ordinal numbers, transfinite induction, and Zorn's Lemma

Hello from Cambridge, Massachusetts! I've just arrived this past week from California, as August 31 marks my last day as a PhD student at Stanford and September 1 my first day as a postdoctoral researcher at MIT. I've been on a significantly reduced work schedule during the move, so rather than working directly on research problems, I've been taking time to bone up some fundamental math skills. This has involved, mostly, an exploration into functional analysis. One of the fundamental theorems of functional analysis is the Hahn-Banach theorem , which is proved using a very standard application of Zorn's lemma . During my reading, though, I realized that I actually had no idea how Zorn's lemma is proved. I've applied it many times in math courses, but the underpinnings of the lemma elude me. Trying to figure out the proof led me down a pretty deep rabbit hole — it turns out Zorn's lemma is basically a simplified packaging of a general technique called "tr