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Notes on the Stability Spectrum

I wrote this document while preparing for my qualifying exam in model theory. The goal was to clarify for myself some of the many equivalent definitions that appear in stability theory, including where countable language assumptions are necessary, and to collect the arguments that go into Shelah’s stability spectrum for countable theories in a compact, readable, and hopefully memorable form. In Section 2, I have attempted to review all of the definitions and facts about these definitions which are used in the remainder and which are specific to stability theory. This was, again, partially done to test myself on this material before the qualifying exam. Section 3 contains four theorems, characterizing stable formulas (1), stable theories (2), superstable theories (3), and totally transcendental theories (4). In Theorems 2, 3, and 4, the last equivalent statement in each serves to drive a hard line between the unstable and the stable theories (2.(3)), the strictly stable and the superstable theories (3.(5)), and the strictly superstable and the totally transcendental theories (4.(3)). Together, these theorems give a nice classification of all countable theories into one of these four types, and this result (the stability spectrum for countable theories) is summed up in the Corollary at the end of the document. But the implications 3.(5) ⇒ (2) and 4.(3) ⇒ (1) only hold in countable theories (as noted in the statements of the theorems). In uncountable theories, the stability spectrum is significantly more complicated (although a version of 3.(5)⇒ (2) still holds, using a cardinal other than ω). Since the equivalence of (4) in Theorem 3 and (2) in Theorem 4 go through these implications, this leaves open the question of whether 3.(4) is equivalent to superstable and whether 4.(2) is equivalent to superstable. Probably these answers are available in Shelah and elsewhere, but I haven’t thought hard about them or gone looking. If you know or find these answers, or if you see any way to improve the arguments given here, or if you find mistakes, please let me know: kruckman@gmail.com.