On the number of 𝐿_{∞𝜔₁}-equivalent non-isomorphic models

Shelah, Saharon; Väisänen, Pauli

doi:10.1090/S0002-9947-00-02604-0

On the number of ${L}_{\infty \omega _1}$-equivalent non-isomorphic models
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by Saharon Shelah and Pauli Väisänen PDF

Trans. Amer. Math. Soc. 353 (2001), 1781-1817 Request permission

Abstract:

We prove that if $\operatorname {ZF}$ is consistent then $\operatorname {ZFC} + \operatorname {GCH}$ is consistent with the following statement: There is for every $k < \omega$ a model of cardinality $\aleph _1$ which is $L_{\infty {\omega _{1}}}$-equivalent to exactly $k$ non-isomorphic models of cardinality $\aleph _1$. In order to get this result we introduce ladder systems and colourings different from the “standard” counterparts, and prove the following purely combinatorial result: For each prime number $p$ and positive integer $m$ it is consistent with $\operatorname {ZFC} + \operatorname {GHC}$ that there is a “good” ladder system having exactly $p^m$ pairwise nonequivalent colourings.

References

Jon Barwise (ed.), The syntax and semantics of infinitary languages, Lecture Notes in Mathematics, No. 72, Springer-Verlag, Berlin-New York, 1968. MR 0234827
J. Barwise and S. Feferman (eds.), Model-theoretic logics, Perspectives in Mathematical Logic, Springer-Verlag, New York, 1985. MR 819531
Paul C. Eklof and Alan H. Mekler, Almost free modules, North-Holland Mathematical Library, vol. 46, North-Holland Publishing Co., Amsterdam, 1990. Set-theoretic methods. MR 1055083
Paul C. Eklof and Saharon Shelah, New nonfree Whitehead groups by coloring, Abelian groups and modules (Colorado Springs, CO, 1995) Lecture Notes in Pure and Appl. Math., vol. 182, Dekker, New York, 1996, pp. 15–22. MR 1415620
Martin Goldstern, Tools for your forcing construction, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 305–360. MR 1234283
E. A. Palyutin, The number of models of $L_{\infty ,\omega _{1}}$-theories, Algebra i Logika 16 (1977), no. 1, 74–87, 124 (Russian). MR 0505220
E. A. Palyutin, The number of models of $L_{\infty ,\omega _{1}}$-theories, Algebra i Logika 16 (1977), no. 1, 74–87, 124 (Russian). MR 0505220
Dana Scott, Logic with denumerably long formulas and finite strings of quantifiers, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 329–341. MR 0200133
Saharon Shelah, Whitehead groups may be not free, even assuming CH. I, Israel J. Math. 28 (1977), no. 3, 193–204. MR 469757, DOI 10.1007/BF02759809
Saharon Shelah, Whitehead groups may not be free, even assuming CH. II, Israel J. Math. 35 (1980), no. 4, 257–285. MR 594332, DOI 10.1007/BF02760652
Saharon Shelah, The consistency of $\textrm {Ext}(G,\,\textbf {Z})=\textbf {Q}$, Israel J. Math. 39 (1981), no. 1-2, 74–82. MR 617291, DOI 10.1007/BF02762854
Saharon Shelah, On the number of nonisomorphic models of cardinality $\lambda \ L_{\infty \lambda }$-equivalent to a fixed model, Notre Dame J. Formal Logic 22 (1981), no. 1, 5–10. MR 603751
Saharon Shelah, On the number of nonisomorphic models in $L_{\infty ,\kappa }$ when $\kappa$ is weakly compact, Notre Dame J. Formal Logic 23 (1982), no. 1, 21–26. MR 634740
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
Saharon Shelah, On the possible number $\textrm {no}(M)=$ the number of nonisomorphic models $L_{\infty ,\lambda }$-equivalent to $M$ of power $\lambda$, for $\lambda$ singular, Notre Dame J. Formal Logic 26 (1985), no. 1, 36–50. MR 766665, DOI 10.1305/ndjfl/1093870759
Saharon Shelah, Around classification theory of models, Lecture Notes in Mathematics, vol. 1182, Springer-Verlag, Berlin, 1986. MR 850051, DOI 10.1007/BFb0098503
Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206, DOI 10.1007/978-3-662-12831-2
Saharon Shelah and Pauli Väisänen, On inverse $\gamma$-systems and the number of ${L}_{\infty ,\lambda }$-equivalent, non-isomorphic models for $\lambda$ singular, J. Symbolic Logic 65 (2000), 272–284.