Abstract
We prove that for λ singular of cofinality κ>ℵ0, if (∀μ<λ)μκ then for some model M, M=(M, R M), R a two place predicate, ‖M‖=λ and no (M)={N/≈:N≡∞, λ M,‖N‖=λ} is quite arbitrary e.g. any μ<λ and λκ (hence 2λ).
See [Sh 5] for the back ground: where the result were proved for M with relations with infinitely many places. By the present paper the only problem left, if we assume V=L, is whether no (M)=λ, may happen for M of cardinality λ for λ singular.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
C.C. Chang, Some remarks on the model theory of infinitary languages:, pp. 36–63 in: The Syntax and Semantics of Infinitary Languages, Lecture Notes in Mathematics, 72, ed., J. Barwise, Springer, 1968.
H.L. Hiller, and S. Shelah, "Singular cohomology in L", Israel Journal of Mathematics. vol. 26 (1977), pp. 313–319.
M. Nadel and J. Stavi, "L ∞λ-equivalence, isomorphism and potential isomorphism," Transactions of the American Mathematical Society, vol. 236 (1978), pp. 51–74.
E. A. Palyutin, "Number of models in L ∞, ω1 theories III," pp. 443–456 in Algebra I Logika, vol. 16, no. 4. (1977): English translation in Algebra and Logic, vol. 16, no. 4 (1977), pp. 299–309.
D. Scott, "Logic with denumerably long formulas and finite strings of quantifiers," pp. 329–341 in The Theory of Models, North Holland, 1965.
S. Shelah, "On the number of non-isomorphic models of cardinality λ,L ∞λ-equivalent to a fixed model", Notre Dame J. of Formal Logic 22 (1981), 5–10.
—, "On the number of non-isomorphic models of power λ ≡⇔λ to a model of power λ, when λ is weakly compact," Notre Dame J. of Formal Logic, 23 (1982), 21–26.
—, "The consistency of Ext (G, ℤ)=ℚ" Israel Journal of Mathematics, 39 (1981), 283–288.
—, "A pair of non-isomorphic ≡∞,λ, Models of Power λ for λ singular with λω = λ". Notre Dame J. of Formal Logics, 25 (1984), 97–104.
—, "On the possible number no (M) = the numbers of non isomorphic models L ∞,λ-equivalent of power λ for λ singular, Notre Dame J. Formal Logic, in press.
Rights and permissions
Copyright information
© 1986 Springer-Verlag
About this chapter
Cite this chapter
Shelah, S. (1986). On the no(M) for M of singular power. In: Around Classification Theory of Models. Lecture Notes in Mathematics, vol 1182. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098507
Download citation
DOI: https://doi.org/10.1007/BFb0098507
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16448-7
Online ISBN: 978-3-540-39788-5
eBook Packages: Springer Book Archive
