On Universality Properties of Reduced Products

Juliette Kennedy & Saharon Shelah

In the early 1970’s S.Tennenbaum proved that all countable models of P A^{−} + ∀_{1} − T h(ℕ) are embeddable into the reduced product ℕ^{ω}/F, where F is the cofinite filter.

Theorem 1 If M is a model of P A^{−} + ∀_{1} − T h(ℕ), and |M| = ℵ_{1}, then M is embeddable into ℕ^{ω}/F.

Theorem 2 If M is a model of P A^{−} + ∀_{1} − T h(ℕ), |M| = λ^{+}, (ℵ_{1}, ℵ_{0}) → (λ^{+}, λ) and F is a regular filter on λ, then M is embeddable into ℕ^{λ}/F

Theorem 3 If F is a regular ultrafilter on λ, (ℵ_{1}, ℵ_{0}) → (λ^{+}, λ) and M is infinite, then M^{λ}/F is λ^{++}-universal. (This answers Open Question 19 of Chang & Keisler, Model Theory, North-Holland).

Addresses Juliette Kennedy: Department of Mathematics, University of Helsinki, Helsinki, Finland

Saharon Shelah : Institute of Mathematics, Hebrew University, Jerusalem, Israel