Complete coinductive theories. II

Lachlan, A. H.

doi:10.1090/S0002-9947-1991-1014253-1

Complete coinductive theories. II
HTML articles powered by AMS MathViewer

by A. H. Lachlan PDF

Trans. Amer. Math. Soc. 328 (1991), 527-562 Request permission

Abstract:

Let $T$ be a complete theory over a relational language which has an axiomatization by $\exists \forall$-sentences. The properties of models of $T$ are studied. It is shown that existential formulas are stable. A theory of forking and independence based on Boolean combinations of existential formulas in $\exists \forall$-saturated models of $T$ is developed for which the independence relation is shown to satisfy a very strong triviality condition. It follows that $T$ is tree-decomposable in the sense of Baldwin and Shelah. It is also shown that if the language is finite, then $T$ has a prime model.

References

J. Baldwin, Definable second-order quantifiers, Model-theoretic logics, Perspect. Math. Logic, Springer, New York, 1985, pp. 445–477. MR 819543
John T. Baldwin and David W. Kueker, Ramsey quantifiers and the finite cover property, Pacific J. Math. 90 (1980), no. 1, 11–19. MR 599315, DOI 10.2140/pjm.1980.90.11
J. T. Baldwin and S. Shelah, Second-order quantifiers and the complexity of theories, Notre Dame J. Formal Logic 26 (1985), no. 3, 229–303. MR 796638, DOI 10.1305/ndjfl/1093870870
G. Cherlin and A. H. Lachlan, Stable finitely homogeneous structures, Trans. Amer. Math. Soc. 296 (1986), no. 2, 815–850. MR 846608, DOI 10.1090/S0002-9947-1986-0846608-8
Victor Harnik and Leo Harrington, Fundamentals of forking, Ann. Pure Appl. Logic 26 (1984), no. 3, 245–286. MR 747686, DOI 10.1016/0168-0072(84)90005-8
I. M. Hodkinson and H. D. Macpherson, Relational structures determined by their finite induced substructures, J. Symbolic Logic 53 (1988), no. 1, 222–230. MR 929387, DOI 10.2307/2274440

Remarks on

stable

categorical theories

A. H. Lachlan, Two conjectures regarding the stability of $\omega$-categorical theories, Fund. Math. 81 (1973/74), no. 2, 133–145. MR 337572, DOI 10.4064/fm-81-2-133-145
A. H. Lachlan, Complete theories with only universal and existential axioms, J. Symbolic Logic 52 (1987), no. 3, 698–711. MR 902985, DOI 10.2307/2274358
A. H. Lachlan, Complete coinductive theories. I, Trans. Amer. Math. Soc. 319 (1990), no. 1, 209–241. MR 957082, DOI 10.1090/S0002-9947-1990-0957082-6
A. H. Lachlan, Some coinductive graphs, Arch. Math. Logic 29 (1990), no. 4, 213–229. MR 1062726, DOI 10.1007/BF01651325
H. D. Macpherson, Graphs determined by their finite induced subgraphs, J. Combin. Theory Ser. B 41 (1986), no. 2, 230–234. MR 859313, DOI 10.1016/0095-8956(86)90046-8
James H. Schmerl, Coinductive $\aleph _0$-categorical theories, J. Symbolic Logic 55 (1990), no. 3, 1130–1137. MR 1071319, DOI 10.2307/2274478
Saharon Shelah, Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 513226