This is a very difficult question that I find myself pondering every so often. I once heard a story that Jim Baumgartner was asked this question at a major logic meeting where all areas — recursion theory, model theory, proof theory, etc. — were well represented and he responded that it was the only area of logic not represented there. Indeed, the best way to describe infinitary combinatorics is to describe what it isn’t, as in the introductory paragraph of Jim’s review of William’s book on the subject:
Combinatorial set theory is frequently distinguished from axiomatic set theory, although the distinction is becoming less and less clear all the time. If there is a difference, it is more one of method than substance. Axiomatic set theory uses the tools of mathematical logic, such as the method of ultrapowers and the theory of forcing and generic sets, while the methods of combinatorial set theory are purely “combinatorial” in nature. In practice, an argument of result is “combinatorial” if it is not overtly model-theoretic, topological, or measure-theoretic.1
I stumbled across the above review while seeking to find a reference for the anecdote about Jim at the meeting. I don’t remember where I got this story from, I may have read it somewhere or I may have heard it from somebody or Jim himself. I didn’t find a reference for the anecdote. However, I did find that Jim wrote some fantastic book reviews. One of the best is his review of Kunen’s classic set theory textbook, where he writes the following about Jech’s encyclopedic Set Theory:
Graduate students went without food to put a copy of this fat green book on their shelves.2
This was the first edition of Jech’s book, of course. The classic is now much fatter, yellow rather than green, and sufficiently affordable that graduate students can acquire a copy without excessive suffering.
- James E. Baumgartner, Review: Neil H. Williams, Combinatorial set theory. Bull. Amer. Math. Soc. (N.S.) 1 (1979), 217–219. [link]
- James E. Baumbarther, Review: Kenneth Kunen, Set theory. An introduction to independence proofs. J. Symb. Logic 51 (1986), 462–464. [link]