nLab Shelah's main gap

Shelah’s main gap is a result in stability theory (part of model theory), which seeks to identify those first-order theories whose models admit some sort of reasonable classification. The ‘main gap’ describes a fundamental dichotomy between theories.

Roughly speaking, a countable first-order theory belongs to one of the two classes:

• (intractable:) has the maximal number of models in all sufficiently large cardinalities

• every model can be decomposed as a tree of countable models

There are some generalizations of this dichotomy beyond first-order theories.

• L. Harrington, M. Makkai, An exposition of Shelah’s “in gap’’: counting uncountable models of ω-stable and superstable theories, Notre Dame J. Formal Logic 26 (1985), no. 2, 139–177 doi euclid