Order Theory for the Iridescent Astronaut
Q and A
Q: Do we really need another order theory tutorial for astronauts? Aren't there enough already?
A: As far as I know, there are no order theory tutorials for astronauts. The title Order Theory for the Iridescent Astronaut was only chosen to get your attention. If this tutorial isn’t for astronauts, who then is it for?
Programmers and computer scientists
Order theory is useful for many forms of reasoning that are often relevant to the tasks programmers perform. Have you ever performed a topological sort? That’s an inherently order theoretic operation, but it’s just the tip of the iceberg. Order theory can also be used for reasoning about hierarchical relations (such as class hierarchies in OOP), algorithms which perform successive refinements of an approximation (such as the binary search), causality in distributed systems, and more.
The mathematically inclined
The goal is to understand order on a deep level, and to do that we’re going to need to work through rigorous proofs. As a prerequisite, you will need familiarity with the following concepts
 Proof by contradiction
 Proof by contrapositive
 Proof by vacuity (i.e. when a claim is vacuously true)
 How to prove two sets are equal
If you don’t know what any of those things are, you may be feeling pretty depressed right now, but you should cheer up. Why? Because you can learn all of these things, starting now! There is a widespread societal problem where people believe they cannot learn anything beyond basic reading, writing, and arithmetic unless they are enrolled in a formal program, working toward an advanced degree. In fact, there is a fairly effective way to learn almost anything:

Step 1: Figure out what the most established and respected introductory textbooks are in the field of your choice. Find out which textbooks universities like Stanford and Princeton are using to teach their introductory courses. (They should be included in publicly visible course syllabi in the courses’ websites.)

Step 2: Choose a textbook, buy it from amazon, read through the chapters and do the exercises.

Step 3: For further info on topics covered in the book, follow the references in the back.
To learn the basics of rigorous mathematics in particular (which includes contrapositive, vacuity, etc.), I recommend Mathematics: A Discrete Introduction by Ed Scheinerman. I read this book in high school. It’s really quite a gentle and approachable, but this sort of thing does not click with everyone.
Begin learning
These tutorials are structured as a nonlinear schema. Start reading here and follow links as you see fit. Or, if you insist on linearity, use the table of contents below, traversing topics from top to bottom.