# How to Make an Introductory Class in Set Theory and Logic Exciting

I am teaching a "proof techniques" class for sophomore math majors. We start out defining sets and what you can do with them (intersection, union, cartesian product, etc.). We then move on to predicate logic and simple proofs using the rules of first order logic. After that we prove simple math statements via direct proof, contrapositive, contradiction, induction, etc. Finally, we end with basic, but important concepts, injective/surjective, cardinality, modular arithmetic, and relations.

I am having a hard time keeping the class interested in the beginning set theory and logic part of the course. It is pretty dry material. What types of games or group activities might be both more enjoyable than my lectures and instructive?

• I hope you don't mind the small alteration I made to the title. I can think of many ways to make a class in set theory and logic exciting (to me at least, and likely Asaf as well). The rudimentary stuff is another story, however. – user642796 Sep 10 '13 at 15:43
• Make them think. Many examples, and then the abstract concept. – Don Larynx Sep 10 '13 at 15:46
• To me, what always interested me, is getting to bottom of things. That alone was a huge motivation for me. Alas most people don't see the beauty in this. – Git Gud Sep 10 '13 at 15:56
• After they learn about tautologies, contradictions and implication, ask them in a way that forces them to answer quickly: $$P\implies \neg P, \text{ tautology or contradiction}?$$ The expected answer is contradiction and, of course, it is neither. This works almost $100\%$ of the time. The shock value should be enough to keep them alert to the fact that intuition is a false friend. – Git Gud Sep 10 '13 at 16:25
• May I humbly suggest my educational, proof-writing software. It addresses the need for immediate feedback, especially for beginners -- they won't have to wonder if they have it right. The tutorial included with it covers most of the topics you list here. Free, full-function download at dcproof.com – Dan Christensen Sep 12 '13 at 14:28

Be excited about sets and logic, and generally what you are talking about.

When I was a freshman I had a TA in calculus 2 that was totally awesome. Not because he was particularly good, and I was particularly uninterested in the topic. But to hear him talk about the theorems was inspiring.

I took from that a lot, and when I was TA'ing intro to logic and set theory, I too tried to be excited about whatever it was that I could be excited about (that is, not the extremely dull theorems, but most of the other things). I would pick exercises which I found exciting, then it was just so much easier to get excited.

The important things are:

1. Keep the class involved. Ask them a questions and wait for them to answer. When my students don't answer I just stare at them and tell them we're not going to continue until they do. I sometimes use an application on my smartphone to make sounds of crickets when they are too quiet, they usually laugh and then they answer.

2. Use examples that you think are awesome. Examples which you think will surprise them. Things they are expecting to be true will bore them, and they won't be sure what there is to be excited about. But if you catch them unprepared then they will have a better chance of enjoying the class.

3. Spice things up with history. Who proved that, peculiar notations from the history of the topic. Don't overdo it, but from time to time it's nice to add some background, especially if people's names are already mentioned.

4. BE EXCITED!

All in all, teaching is much like story telling. You tell a story, and they listen and learn from it. If you think that the story is dull and uninteresting, then your crowd will think so as well.

• +5 for all this! – Peter Smith Sep 10 '13 at 16:22
• I am sorry that I forgot to reward you for your answer. The bounty placed on the question has kindly reminded me. I am happy to give you the points. Your participation on this site has helped me too many times to count. – Joe Johnson 126 Jul 10 '17 at 1:58
• @Joe: Don't worry about it. And thanks! :) – Asaf Karagila Jul 10 '17 at 4:38

Can I echo @dfeur's suggestion that a bit of history and conceptual commentary could be intriguing/fun/motivational (at least for more intellectually curious students)?

Sets How did sets get into the story in the nineteenth century (the arithmetization of analysis)? Frege's disaster and Russell's paradox. Zermelo's response. The idea of the cumulative hierarchy and other conceptions of the universe of sets. Why such a simple claim as the Continuum Hypothesis remains problematic.

Logic Something about how/why classical first-order logic becomes standard. Why constuctivists balk at excluded middle. Why it is so difficult to do better than the material conditional to regiment indicative conditionals. The motivation for Frege's quantifier/variable regimentation of general propositions. The conceptual motivations behind different approaches to logic (axiomatic, natural deduction). Whether some mathematical reasoning is not first-order.

It's good for students to see e.g. that while (versions of) first-order logic as a theory are of course cleanly definable, it isn't so cut and dried why we've come to treat FOL as canonical. And as @Asaf says, if you find [some of] these questions intriguing, then your puzzlement and interest in them should be infectious.

Sometimes it makes sense to teach a little bit backwards. Rather than always teaching the foundations first and then building on top of them, it sometimes pays to build a little higher-level context first, and then build foundations underneath. One way is to use a partially historical approach. That is, start by teaching about the history of an idea, and a little of the mathematics and/or philosophy surrounding that history, before you actually teach the idea.

maybe start with some riddles from http://en.wikipedia.org/wiki/Charles_Lutwidge_Dodgson or http://en.wikipedia.org/wiki/Raymond_Smullyan both have made books full with logical riddles