Foreword

This book has been reviewed pretty thoroughly already. Rather than restate each chapter, I'll be sharing insights: some book-specific, some general.

I am quite proud of my time-to-completion for this book - just over a week, working around a very strenuous courseload. I went from having to focus really hard to pick up new concepts to reading notation nearly as fluently as the English surrounding it. The chapters started making sense - it felt less random, and more like a coherent story wherein the protagonist slowly adds $P$ owers to their arsenal.

Naïve Set Theory

Functions

Functions $f : X \to Y$ are just static sets of ordered pairs ${(x, f (x)) : x \in X}$ . They are not dynamic indexing functions, they do not perform efficient lookup, please do not waste an hour of your life trying to figure out how you could do such a thing within the simple machinery afforded by set theory up to that point.

This is one of those things that Nate talked about - how skipping over just one word a few chapters prior can cause you to waste hours. During my confusion, I knew this was probably the case, but I still couldn't manage to immediately overcome my intuitions of what a function should be. This is one reason I'm open to working through the MIRI Research Guide with others.

Families

Families are, ironically enough, just a special kind of function; don't let your intuition fool you - they aren't "groups of functions". A family belonging to $\prod_{i \in I} X_{i}$ maps each element $i$ of the index set $I$ to an element $x_{i} \in X_{i}$ . For example, a family from ${1, 2}$ to $X_{1} := {c a t, d o g}, X_{2} := {t e n t, fire}$ could be ${(1, c a t), (2, fire)}$ (thanks to Dacyn for helping me clarify my writing).

Zorn's Lemma

I spent three hours staring at this proof. I understood what ZL meant. I grasped the relevant concepts. I read other versions of the proof. I still spent three long hours on this damn proof, and then I went to my classes. I don't know why I ended up figuring it out, but I suspect it was a combination of two factors: my brain worked through some things during the day, and I really wanted it. On the bus home, I mentally snapped and decided I was going to understand the proof. And I did.

I'm pleased to share my detailed proof outline of Zorn's Lemma, the product of many hours of ambient exasperation, rewritten in my own notation. Looking back, the proof in the book was pretty bad; it ... ,,,,,,,,,