Pseudo-Finite Homogeneity and Saturation

When analyzing database query languages a property of theories, the pseudo-finite homogeneity property has been introduced and applied (cf. [3]). We show that a stable theory has the pseudo-finite homogeneity property just in case its expressive power for finite states is bounded. Moreover, we introduce the corresponding pseudo-finite saturation property and show that a theory fails to have the finite cover property iff it has the pseudo-finite saturation property. 1 Pseudo-finite homogeneity Throughout let T be a complete first-order theory in a countable language L with infinite models. Suppose that ρ is a finite non-empty set of relation symbols not contained in L. Set L(ρ) := L ∪ ρ. If M is a model of T , and (M, P̄ ) is an L(ρ)-structure with, say, P̄ = P1 . . . Pr, then P̄ is a (ρ)-state in M . fld(P̄ ), the field or active domain of the state P̄ , is the set fld(P1)∪ . . .∪ fld(Pr), where fld(Pj) is the field of the relation Pj . P̄ is a finite state, if every fld(Pj) is finite and non-empty. In the following we will denote finite states by s̄, s̄′ . . .. A state P̄ in M is pseudo-finite, if (M, P̄ ) is a model of F (T, ρ), the theory of all finite states, i.e., F (T, ρ) := Th({(N, s̄) | N |= T, (N, s̄) an L(ρ)-structure, s̄ a finite state}). In general, we use r̄, r̄′, t̄ . . . to denote pseudo-finite states. Example 1.1 For L := ∅, T the L-theory of infinite sets, and ρ := {P} with unary P , a subset r of a model M of T is pseudo-finite iff M \ r is infinite. In fact, for every k, the complement of every finite subset contains at least k elements, hence the same holds for a pseudo-finite subset. If M \ r and r are infinite and l ≥ 1, then (M, r) satisfies the same sentences of quantifier rank ≤ l as (M, s), where |s| = l. Therefore, (M, r) |= F (T, ρ). We collect some properties of pseudo-finite states: ∗e-mail address: {flum, ziegler}@uni-freiburg.de