[Mahratta]

This course was, as advertised, an introduction to modern mathematical logic, and particularly to basic proof theory and model theory. So, what's the point?

Mathematics is defined by its insistence on formal proof, given some set of standards (a formal system -- axioms and rules of inference).

At the beginning of the 20th century, mathematicians began asking questions like "given any formal system, set of assumptions (maybe empty) and conclusion, is there an algorithm which will output whether or not the conclusion is provable in the system from the assumptions?" (this is highly linked to the foundations of computer science).

Logicians then had two apparently different tasks: (i) Proof theory: to study systems of logical formulas and sentences and provability of sentences from such systems (ii) Model theory: to study models -- i.e. mathematical structures (or relational structures in computer science, even linguistic structures) of these sets of formulas. Goedel proved by the Completeness theorem that a formal proof system is consistent if and only if it has a model, connecting the two disciplines.

That's the first half of the course. Lots of proofs, can be somewhat computational. But it gets better as you go along, and the profs try to give plenty of motivating mathematical examples.

You then start getting into model theory, the interesting stuff. You cover the compactness theorem -- which essentially states that, topologically, the space of well-formed formulas is compact, the Loewenheim-Skolem theorems, which allow each theory with one infinite model to have a model of any infinite size, and more.

I had Dr Speissegger, he's an o-minimalist (a particular branch of model theory which deals with structures that are topologically like the real numbers) and was an excellent professor, possibly the best I've had. He uses a good combination of overheads and chalkboard proofs, and allows the class to participate a lot.