# Set Up for Success: Insights from 'Naïve Set Theory'

Foreword
========

[This book](http://smile.amazon.com/Naive-Set-Theory-Paul-Halmos/dp/1614271313/) has been reviewed [pretty](https://www.lesserwrong.com/posts/Ee8CZW7wzaNdCENYG/book-review-naive-set-theory-miri-course-list) [thoroughly](https://www.lesserwrong.com/posts/FvA2qL6ChCbyi5Axk/book-review-naive-set-theory-miri-research-guide) 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 < refresh to render LaTeX > owers to their arsenal.

Naïve Set Theory
================

Functions
---------

Functions < refresh to render LaTeX > are _just_ **static sets** of ordered pairs < refresh to render LaTeX > . 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](https://www.lesserwrong.com/posts/uX3HjXo6BWos3Zgy5/the-mechanics-of-my-recent-productivity) \- 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](https://intelligence.org/research-guide/#two) 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 < refresh to render LaTeX > maps each element < refresh to render LaTeX > of the index set < refresh to render LaTeX > to an element < refresh to render LaTeX > . For example, a family from < refresh to render LaTeX > to < refresh to render LaTeX > could be < refresh to render LaTeX > (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](https://www.overleaf.com/read/ppftcthcvjxs) 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 was neither succinct nor intuitive, but instead imposed a marais of mental variable tracking on the reader. I think mine is at least a little better, if not fully fleshed-out at all junctures.

Proof Calibration
-----------------

As someone without a formal degree in mathematics, it was important for me to monitor how I approached the exercises in the book. Whenever the author began a proof, I tried to generate a mental proof sketch before reading further. Sometimes, I thought the proof would be easy and short, but it would turn out that my approach wasn’t rigorous enough. This was valuable feedback for calibration, and I intend to continue this practice. I'm still worried that down the line and in the absence of teachers, I may believe that I've learnt the research guide with the necessary rigor, go to a MIRIx workshop, and realize I hadn't been holding myself to a sufficiently high standard. Suggestions for ameliorating this would be welcome.

Forwards
========

Anticipation
------------

One factor which helped me succeed was that I ensured my morning reading was what I most looked forward to each day. I was excited to go to sleep, wake up early, prepare a delicious breakfast, and curl up near the fireplace with book and paper handy. Trivial inconveniences can be fatal - do whatever you must to ensure you properly respect and anticipate your study time.

Defense with the Dark Arts
--------------------------

[The most useful productivity-related advice I ever read](http://mindingourway.com/dark-arts-of-rationality/) was by Nate Soares (**Dark Arts warning**), and it relates to imbuing your instrumental goals with terminal values. Ever since having read that advice, every tedious assignment, every daily routine, every keystroke - they're all backed up by an intense desire to _do something_ about the precarious situation in which humanity finds itself.

Internal Light
--------------

If you don't know where to start, I think the internal fire has to be lit first - don't try to force yourself to do this (or anything else) because you [should](http://mindingourway.com/not-because-you-should/). [Stop the guilt-based motivation](http://mindingourway.com/guilt/), proudly stake out what you want, and transform your life into a dazzling assortment of activities and tasks imbued with your terminal values, your brightest visions for yourself and the future.