On the number of ${L}_{\infty \omega _1}$-equivalent non-isomorphic models
Authors:
Saharon Shelah and Pauli VΓ€isΓ€nen
Journal:
Trans. Amer. Math. Soc. 353 (2001), 1781-1817
MSC (2000):
Primary 03C55; Secondary 03C75, 03E05
DOI:
https://doi.org/10.1090/S0002-9947-00-02604-0
Published electronically:
December 29, 2000
MathSciNet review:
1707477
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 βstandardβ 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 βgoodβ ladder system having exactly $p^m$ pairwise nonequivalent colourings.
- 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. 15β22. 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. 305β360. MR 1234283
- E. A. Palyutin, The number of models of $L_{\infty ,\omega _{1}}$-theories, Algebra i Logika 16 (1977), no. 1, 74β87, 124 (Russian). MR 0505220
- E. A. Palyutin, The number of models of $L_{\infty ,\omega _{1}}$-theories, Algebra i Logika 16 (1977), no. 1, 74β87, 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. 329β341. MR 0200133
- Saharon Shelah, Whitehead groups may be not free, even assuming CH. I, Israel J. Math. 28 (1977), no. 3, 193β204. MR 469757, DOI https://doi.org/10.1007/BF02759809
- Saharon Shelah, Whitehead groups may not be free, even assuming CH. II, Israel J. Math. 35 (1980), no. 4, 257β285. MR 594332, DOI https://doi.org/10.1007/BF02760652
- Saharon Shelah, The consistency of ${\rm Ext}(G,\,{\bf Z})={\bf Q}$, Israel J. Math. 39 (1981), no. 1-2, 74β82. MR 617291, DOI https://doi.org/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, 5β10. 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, 21β26. 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 ${\rm no}(M)=$ 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, 36β50. MR 766665, DOI https://doi.org/10.1305/ndjfl/1093870759
- Saharon Shelah, Around classification theory of models, Lecture Notes in Mathematics, vol. 1182, Springer-Verlag, Berlin, 1986. MR 850051
- Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206
- Saharon Shelah and Pauli VΓ€isΓ€nen, On inverse $\gamma$-systems and the number of ${L}_{\infty ,\lambda }$-equivalent, non-isomorphic models for $\lambda$ singular, J. Symbolic Logic 65 (2000), 272β284.
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03C55, 03C75, 03E05
Retrieve articles in all journals with MSC (2000): 03C55, 03C75, 03E05
Additional Information
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel and Rutgers University, Hill Ctr-Busch, New Brunswick, New Jersey 08903
MR Author ID:
160185
ORCID:
0000-0003-0462-3152
Email:
shelah@math.huji.ac.il
Pauli VΓ€isΓ€nen
Affiliation:
Department of Mathematics, P.O. Box 4, 00014 University of Helsinki, Finland
Email:
pauli.vaisanen@helsinki.fi
Keywords:
Number of models,
ladder system,
uniformization,
infinitary logic,
iterated forcing
Received by editor(s):
April 28, 1997
Published electronically:
December 29, 2000
Additional Notes:
The first author thanks GIF for its support of this research, and also the University of Helsinki for funding a visit of the first author to Helsinki in August 1996. This is his paper number 646.
This paper is the second authorβs Licentiateβs thesis. The second author did his share of the paper under the supervision of Tapani Hyttinen.
Article copyright:
© Copyright 2000
American Mathematical Society
