Source: J. Symbolic Logic Volume 74, Issue 3
(2009), 914-928.
We study when classes can have the disjoint amalgamation property
for a proper initial segment of cardinals.
Theorem A
For every natural number k, there is a class Kk defined by a
sentence in Lω₁,ω that has no models of
cardinality greater than ℶk+1, but Kk has the
disjoint amalgamation property on models of cardinality less than
or equal to ℵk-3 and has models of cardinality ℵ{k}-1.
More strongly, we can have
disjoint amalgamation up to ℵα for
α < ω₁, but have a bound on size of
models.
Theorem B
For every countable ordinal α, there is a class Kα
defined by a sentence in Lω₁,ω that has no models
of cardinality greater than ℶω₁, but K does
have the disjoint amalgamation property on models of cardinality
less than or equal to ℵα.
Finally we show that we can extend the ℵα to
ℶα in the second theorem consistently with ZFC and while having
ℵi≪ ℶi for 0< i≤ α. Similar results hold for
arbitrary ordinals α with |α|=κ and Lκ⁺,ω.
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription.
Read more about accessing full-text
References
John T. Baldwin, Categoricity, www.math.uic.edu/~jbaldwin.
Rami Grossberg, Classification theory for non-elementary classes, Logic and algebra (Yi Zhang, editor), Contemporary Mathematics, vol. 302, American Mathematical Society, 2002, pp. 165--204.
Rami Grossberg and M. Van Dieren, Shelah's categoricity conjecture from a successor for tame abstract elementary classes, Journal of Symbolic Logic, vol. 71 (2006), pp. 553--568.
Greg Hjorth, Knight's model, its automorphism group, and characterizing the uncountable cardinals, Notre Dame Journal of Formal Logic, (2007).
J.F. Knight, A complete $l_\omega_1,\omega$-sentence characterizing $\aleph_1$, Journal of Symbolic Logic, vol. 42 (1977), pp. 151--161.
Mathematical Reviews (MathSciNet):
MR491141
Michael C. Laskowski and Saharon Shelah, On the existence of atomic models, Journal of Symbolic Logic, vol. 58 (1993), pp. 1189--1194.
Olivier Lessmann, Upward categoricity from a successor cardinal for an abstract elementary class with amalgamation, Journal of Symbolic Logic, vol. 70 (2005), no. 2, pp. 639--660.
M. Morley, Omitting classes of elements, The theory of models (Addison, Henkin, and Tarski, editors), North-Holland, Amsterdam, 1965, pp. 265--273.
Mathematical Reviews (MathSciNet):
MR201305
Saharon Shelah, Classification theory and the number of nonisomorphic models, North-Holland, 1978.
Mathematical Reviews (MathSciNet):
MR513226
--------, A weak generalization of MA to higher cardinals, Israel Journal of Mathematics, vol. 30 (1978), pp. 297--306.
Mathematical Reviews (MathSciNet):
MR505492
--------, Simple unstable theories, Annals of Mathematical Logic, vol. 19 (1980), pp. 177--203.
Mathematical Reviews (MathSciNet):
MR595012
--------, Classification theory for nonelementary classes. I. The number of uncountable models of $\psi \in L_\omega _1\omega $, part A, Israel Journal of Mathematics, vol. 46 (1983), no. 3, pp. 212--240, paper 87a.
Mathematical Reviews (MathSciNet):
MR733351
--------, Classification theory for nonelementary classes. I. The number of uncountable models of $\psi \in L_\omega _1\omega $, part B, Israel Journal of Mathematics, vol. 46 (1983), no. 3, pp. 241--271, paper 87b.
Mathematical Reviews (MathSciNet):
MR730343
--------, Categoricity for abstract classes with amalgamation, Annals of Pure and Applied Logic, vol. 98 (1999), pp. 261--294, paper 394. Consult Shelah for post-publication revisions.
Ioannis Souldatos, Notes on cardinals that characterizable by a complete (Scott) sentence, preprint.
