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The number of pairwise non-elementarily-embeddable models

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah*
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel Department of Mathematics, Simon Fraser University, Burnaby, British Columbia B5A 1S6, Canada Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Abstract

We get consistency results on I(λ, T, T) under the assumption that D(T) has cardinality > ∣T∣. We get positive results and consistency results on IE(λ, T 1, T).

The interest is model-theoretic, but the content is mostly set-theoretic: in Theorems 1–3, combinatorial; in Theorems 4–7 and 11(2), to prove consistency of counterexamples we concentrate on forcing arguments; and in Theorems 8–10 and 11(1), combinatorics for counterexamples; the rest are discussion and problems. In particular:

(A) By Theorems 1 and 2, if TT 1 are first order countable, T complete stable but ℵ0-unstable, λ > ℵ0, and ∣D(T)∣ > ℵ0, then IE(λ, T 1, T) > Min{2 λ , ℶ2}.

(B) By Theorems 4,5,6 of this paper, if e.g. V = L, then in some generic extension of V not collapsing cardinals, for some first order TT 1, ∣T∣ = ℵ0, ∣T 1∣ = ℵ1, ∣D(T)∣ = ℵ2 and IE(ℵ2, T1, T) = 1.

This paper (specifically the ZFC results) is continued in the very interesting work of Baldwin on diversity classes [Bl]. Some more advances can be found in the new version of [Sh300] (see Chapter III, mainly §7); they confirm 0.1, 0.2 and 14(1), 14(2).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1989

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Footnotes

1

This work was partially supported by grant number A1030 from the Natural Sciences and Engineering Research Council of Canada, and also by grants from the U.S. National Science Foundation and the U.S.-Israel Binational Science Foundation. This paper is number 262 in the list on pp. 398–418 of [Sh300].

2

I thank Rami Grossberg for many corrections, and in particular for rewriting the proof of Theorem 1; and also John Baldwin for adding explanations. The paper's volume has grown considerably under their pressure. I also thank Alice Leonhardt for the beautiful typing of the manuscript.

References

REFERENCES

[B] Baumgartner, J., personal communication.Google Scholar
[B1] Baldwin, J., Diverse classes, this Journal, vol. 54 (1989), pp. 875893.Google Scholar
[EM] Ehrenfeucht, A. and Mostowski, A., Models of axiomatic theories admitting automorphisms, Fundamenta Mathematicae, vol. 43 (1956), pp. 5068.CrossRefGoogle Scholar
[GH] Galvin, F. and Hajnal, A., Inequalities for cardinal powers, Annals of Mathematics, ser. 2, vol. 101 (1975), pp. 491498.CrossRefGoogle Scholar
[K] Keisler, H. J., Logic with the quantifier “there exist uncountably many”, Annals of Mathematical Logic, vol. 1 (1970), pp. 193.CrossRefGoogle Scholar
[Lv] Laver, R., Making the supercompactness of κ indestructible under κ-directed closed forcings, Israel Journal of Mathematics, vol. 29 (1978), pp. 385388.CrossRefGoogle Scholar
[Mg1] Magidor, M., Changing of cofinalities of cardinals, Fundamenta Mathematicae, vol. 99 (1978), pp. 6171.CrossRefGoogle Scholar
[Mg2] Magidor, M., A covering theorem, preprint.Google Scholar
[Mi] Mitchell, W. J., Aronszajn trees and the independence of the transfer property, Annals of Mathematical Logic, vol. 5 (1972), pp. 2146.CrossRefGoogle Scholar
[RSh117] Rubin, M. and Shelah, S., Combinatorial problems on trees: partitions, ⊿-systems and large free subsets, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 4382.CrossRefGoogle Scholar
[ShA1] Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[ShA1a] Shelah, S., Classification theory: completed for countable theories, North-Holland, Amsterdam (in press).Google Scholar
[ShA2] Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[Sh80] Shelah, S., A weak generalization of MA to higher cardinals, Israel Journal of Mathematics, vol. 30 (1978), pp. 297306.CrossRefGoogle Scholar
[Sh100] Shelah, S., Independence results, this Journal, vol. 45 (1980), pp. 563573.Google Scholar
[Sh111] Shelah, S., On power of singular cardinals, Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 263299.CrossRefGoogle Scholar
[Sh136] Shelah, S., Constructions of many complicated uncountable structures and Boolean algebras, Israel Journal of Mathematics, vol. 45 (1983), pp. 100146.CrossRefGoogle Scholar
[Sh175] Shelah, S., On universal graphs without instances of CH, Annals of Pure and Applied Logic, vol. 26(1984), pp. 7587.CrossRefGoogle Scholar
[Sh175a] Shelah, S., Universal graphs without instances of CH, revisited, Israel Journal of Mathematics (to appear).Google Scholar
[Sh272] Shelah, S., On almost categorical theories, Classification theory (proceedings, Chicago, 1985), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin, 1988, pp. 488500.Google Scholar
[Sh282] Shelah, S., Successor of singulars, productivity of chain conditions and cofinalities of reduced products of cardinals, Israel Journal of Mathematics, vol. 62 (1988), pp. 213256.CrossRefGoogle Scholar
[Sh300] Shelah, S., Universal classes, chapters I–IV, Classification theory (proceedings, Chicago, 1985), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin, 1988, pp. 264418.CrossRefGoogle Scholar
[Sh345] Shelah, S., Products of regular cardinals and cardinal invariants of Boolean algebras, Israel Journal of Mathematics (in press).Google Scholar
[Sh355] Shelah, S., ℵω+1 has a Jónsson algebra, preprint.Google Scholar
[Si] Silver, J., Some applications of model theory in set theory, Annals of Mathematical Logic, vol. 3 (1971), pp. 45110.CrossRefGoogle Scholar