Wednesday, January 07, 2009

Do the proof

In Hodges's model theory book, there is a proof that sort of surprised me. The proof is for lemma 9.1.5 and runs as follows.

A Skolem theory is axiomatized by a set of ∀1 sentences and modulo the theory, every formula is equivalent to a quantifier-free formula. Now quote Lemma 9.1.3 and Theorem 9.1.1.

What surprised me is the last sentence. It is more like a recipe, telling you what to do. Some other proofs in the book have this character to a small degree, but this one stood out a little. The proof is fine, but, unlike many other proofs, this one seems to encourage the reader to do the proof as well. Is this sort of "more active" style of writing proofs common? I don't think I've come across it much at all in the things I read. I'd expect the last line of the proof to go: "The result then follows from Lemma 9.1.3 and Theorem 9.1.1."

2 comments:

Bryan said...

Ha! A scavenger hunt proof. Would this proof of the isoperimetric theorem count too?

1.Go get a cat.
2. Put it in a cold room, and wait until it falls asleep.
3. Observe the approximate three-dimensional shape that the cat assumes (i.e., a sphere).
4. If the cat assumes this shape, it is because that shape is the warmest in three dimensions.
5. The warmest shape is the one that minimizes the ratio of surface area to volume.
6. Therefore: the sphere is the solid that minimizes the ratio of surface area to volume in three dimensions.

(More discussion here.)

Shawn said...

It would. I'd also love to see a proof in print that began "go get a cat."