> I like geometry. Let's start with Hilbert's axioms > for Euclidean > geometry and prove a couple theorems and a couple > statements that > look like they are theorems but are demonstrably > false. >
Good luck. It has been my experience that even the most trivial geometric proofs starting from formal axioms are very long and tedious indeed. I cannot recommend it for the beginner. Better to start with the simplest possible systems, e.g. logic and set theory that are the basis for EVERY branch of mathematics.