Hostname: page-component-5d59c44645-dknvm Total loading time: 0 Render date: 2024-02-28T00:54:26.427Z Has data issue: false hasContentIssue false

Transfering saturation, the finite cover property, and stability

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin
Department of Mathematics, University of Illinois at Chicago, Chicago, IL 60680., USA E-mail:
Rami Grossberg
Department of Mathematics, Carnegie Mellon University, Pittsburgh. Pa 15213, USA E-mail:
Saharon Shelah
Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91094, Israel Department of Mathematics, Rutgers University, New Brunswick. NJ 08902, USA E-mail:


Saturation is (μ, κ)-transferable in T if and only if there is an expansion T1 of T with |T1| = |T| such that if M is a μ-saturated model of T1 and |M| ≥ κ then the reduct M|L(T) is κ-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (ℵ0, λ)-transferable or (κ(T), λ)-transferable for all λ. Further if for some μ ≥ |T|,2μ > μ+, stability is equivalent to for all μ ≥ |T|, saturation is (μ, 2μ)-transferable.

Research Article
Copyright © Association for Symbolic Logic 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



[Ca] Casanovas, Enrique, Compactly expandable models and stability, this Journal, vol. 60 (1995), pp. 673–683.Google Scholar
[CK] Chang, C.C. and Keisler, H. Jerome, Model Theory, North-Holland Publ. Co., 1990.Google Scholar
[Ke] Keisler, H. Jerome, Ultraproducts which are not saturated, this Journal, vol. 32 (1967), pp. 23–46.Google Scholar
[Ku] Kunen, Kenneth, Ultrafilters and independent sets, Transactions of the American Mathematical Society, vol. 172 (1972), pp. 199–206.Google Scholar
[Sh:10] Shelah, Saharon, Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory, Annals of Mathematical Logic, vol. 3 (1971), pp. 271–362, —MR: 47:6475, (02H05).Google Scholar
[Sh:c] Shelah, Saharon, Classification Theory and the Number of Nonisomorphic Models, rev. ed., North-Holland, Amsterdam, 1990.Google Scholar