Abstract
The logic \({\mathbb{L}}_\theta ^1\) was introduced in [She12]; it is the maximal logic below \({{\mathbb{L}}_{\theta, \theta}}\) in which a well ordering is not definable. We investigate it for θ a compact cardinal. We prove that it satisfies several parallels of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are \({\mathbb{L}}_\theta ^1\)-equivalent iff for some ω-sequence of θ-complete ultrafilters, the iterated ultrapowers by it of those two models are isomorphic.
Also for strong limit λ>θ of cofinality \({\aleph _0}\), every complete \({\mathbb{L}}_\theta ^1\)-theory has a so-called special model of cardinality λ, a parallel of saturated. For first order theory T and singular strong limit cardinal λ, T has a so-called special model of cardinality λ. Using “special” in our context is justified by: it is unique (fixing T and λ), all reducts of a special model are special too, so we have another proof of interpolation in this case.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. C. Chang and H. J. Keisler, Model Theory, Studies in Logic and the Foundation of Mathematics, Vol. 73, North–Holland, Amsterdam, 1973.
A. Ehrenfeucht and A. Mostowski, Models of axiomatic theories admitting automorphisms, Fundamenta Mathematicae 43 (1956), 50–68.
H. Gaifman, Elementary embeddings of models of set-theory and certain subtheories, in Axiomatic Set Theory, Part II, Proceedings of Symposia in Pure Mathematics, Vol. 13, American Mathematical Society, Providence, RI, 1974, pp. 33–101.
W. Hodges and S. Shelah, Infinite games and reduced products, Annals of Mathematical Logic 20 (1981), 77–108.
H. J. Keisler, Ultraproducts and elementary classes, Indagationes Mathematicae 64 (1961), 477–495.
H. J. Keisler, Limit ultrapowers, Transactions of the American Mathematical Society 107 (1963), 382–408.
S. Kochen, Ultraproducts in the theory of models, Annals of Mathematics 74 (1961), 221–61.
J. A. Makowsky, Compactness, embeddings and definability, in Model-Theoretic Logics, Perspectives in Mathematical Logic, Springer, New York, 1985, pp. 645–716.
S. Shelah, Model theory for a compact cardinal, https://arxiv.org/abs/1303.5247.
S. Shelah, Every two elementarily equivalent models have isomorphic ultrapowers, Israel Journal of Mathematics 10 (1971), 224–233.
S. Shelah, Nice infinitary logics, Journal of the American Mathematical Society 25 (2012), 395–427.
S. Shelah, A.E.C. with not too many models, in Logic Without Borders, Ontos Mathematical Logic, Vol. 5, de Gruyter, Berlin–Boston, MA, 2015, pp. 367–402.
R. M. Solovay, Strongly compact cardinals and the gch, in Proceedings of the Tarski Symposium, Berkeley 1971, Proceedings of Symposia in Pure Mathematics, Vol. 25, American Mathematical Society, Providence, RI, 1974, pp. 365–372.
J. Väänänen, Models and Games, Cambridge Studies in Advanced Mathematics, Vol. 132, Cambridge University Press, Cambridge, 2011.
Acknowledgment
The author thanks Alice Leonhardt for the beautiful typing. We thank the referee for many helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author would like to thank the Israel Science Foundation for partial support of this research (Grant No. 1053/11). References like [She12, 2.11=La18] means we cite from [She12], Claim 2.11 which has label La18; this helps if [She12] will be revised.
This paper was separated from [She] which was first typed May 10, 2012; so was IJM 7367. This is the author’s paper no. 1101.
Rights and permissions
About this article
Cite this article
Shelah, S. Isomorphic limit ultrapowers for infinitary logic. Isr. J. Math. 246, 21–46 (2021). https://doi.org/10.1007/s11856-021-2226-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-021-2226-x