Years ago, when I was a student and young professor in Berkeley, one often heard it said that the subject of logic or at least metamathematics, according to the Tarski school, could be divided into four subjects: model theory, set theory, computability theory (then called recursion theory) and proof theory.
I have long felt that this taxonomy has become increasingly inadequate as a description, and so at the start of my course Logic for Philosophers yesterday, I tried to draw a somewhat fuller picture on the whiteboard. You can see my diagram above, showing the main parts of logic, occupying regions both within mathematics, within philosophy and within computer science, as well as filling out regions that might be considered between these subjects.
OK, this picture is a first attempt, and I’ll try to improve it. Please post comments and criticisms below.
It would certainly be possible to flesh out subareas of nearly all the subjects mentioned. We may imagine set theory, for example, broken into descriptive set theory, Borel equivalence relation theory, large cardinals, forcing, cardinal characteristics, and so on; and similarly we may break up the huge subjects of model theory, computability theory and proof theory. In particular, most subjects don’t fit neatly into a small region, since almost all the subjects have parts that touch areas very far away. Computability theory, for example, touches not only complexity theory and computer science, but also model theory in computable model theory, as well as reverse mathematics, foundations of mathematics, proof theory and so on. Category theory is a kind of diffuse superposition onto the entire diagram, with comparatively less direct interaction with these other areas. Proof theory should be closer to reverse mathematics and to model theory, and probably closer to mathematics.