DISCOURAGING RESULTS FOR ULTRAIMAGINARY INDEPENDENCE THEORY

DISCOURAGING RESULTS FOR ULTRAIMAGINARY INDEPENDENCE THEORY ITAY BEN-YAACOV Abstract. Dividing independence for ultraimaginaries is neither symmetric nor transitive. Moreover, any notion of independence satisfying certain axioms (weaker than those for independence in a simple theory) and defined for all ultraimaginary sorts, is nec- essarily trivial. Introduction Assume that we work in a first order simple theory (see [Wag00] for a general expo- sition). Then dividing, or rather non-dividing, defines a ternary independence relation |^ on possibly infinite tuples, satisfying: Invariance: a |^ c b depends solely on tp(a, b, c). Symmetry: a |^ c b ?? b |^ c a. Transitivity: a |^ c bd ?? a |^ c b ? a |^ bc d. Monotonicity: If a |^ c b and b ? ? dcl(b) then a |^ c b ?. Finite character: a |^ c b if and only if a ? |^ c b ? for every finite sub-tuples a? ? a, b? ? b. Extension: For every a, b, c there is a? ?c a such that a? |^ c b. In fact, |^ satisfies two additional properties, namely the local character and the independence theorem, with which we do not deal here.

• then


• over ?


• every finite


• r1 a¯?


• bf cg


• dividing independence


• indiscernible sequence