Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A new Löwenheim-Skolem theorem

Authors: Matthew Foreman and Stevo Todorcevic
Journal: Trans. Amer. Math. Soc. 357 (2005), 1693-1715
MSC (2000): Primary 03C55
Published electronically: December 16, 2004
MathSciNet review: 2115072
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper establishes a refinement of the classical Löwenheim-Skolem theorem. The main result shows that any first order structure has a countable elementary substructure with strong second order properties. Several consequences for Singular Cardinals Combinatorics are deduced from this.

References [Enhancements On Off] (What's this?)

  • 1. Lev Bukovský. Changing cofinality of $\aleph \sb{2}$. In Set theory and hierarchy theory (Proc. Second Conf., Bierutowice, 1975), pp. 37-49. Lecture Notes in Math., Vol. 537. Springer, Berlin, 1976. MR 55:5441
  • 2. Lev Bukovský and Eva Copláková-Hartová. Minimal collapsing extensions of models of ${\rm {z}{f}{c}}$. Ann. Pure Appl. Logic, 46(3):265-298, 1990. MR 92e:03077
  • 3. Maxim R. Burke and Menachem Magidor. Shelah's ${\rm PCF}$ theory and its applications. Ann. Pure Appl. Logic, 50(3):207-254, 1990. MR 92f:03053
  • 4. James Cummings, Matthew Foreman, and Menachem Magidor. Squares, scales and stationary reflection. J. Math. Log. 1(1):35-98, 2001. MR 2003a:03068
  • 5. James Cummings, Matthew Foreman, and Menachem Magidor. Canonical structure in the universe of set theory, parts I and II. To Appear in J. Pure Appl. Logic.
  • 6. M. Foreman, M. Magidor, and S. Shelah. Martin's maximum, saturated ideals, and nonregular ultrafilters. I. Ann. of Math. (2), 127(1):1-47, 1988. MR 89f:03043
  • 7. Matthew Foreman. Stationary sets, Chang's conjecture and Partition theory. In Set Theory: The Hajnal Conference. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 58, Amer. Math. Soc., Providence, RI, 2002, pp. 73-94. MR 2003e:03089
  • 8. Matthew Foreman and Menachem Magidor. Mutually stationary sets and the saturation of the non-stationary ideal on $P_\kappa(\lambda)$. Acta Math., 186(2):271-300, 2001. MR 2002g:03094
  • 9. Matthew Foreman and Menachem Magidor. Large cardinals and definable counterexamples to the continuum hypothesis. Ann. Pure Appl. Logic, 76(1):47-97, 1995. MR 96k:03124
  • 10. Matthew Foreman and Menachem Magidor. A very weak square principle. J. Symbolic Logic, 62(1):175-196, 1997. MR 98i:03062
  • 11. Fred Galvin and András Hajnal. Inequalities for cardinal powers. Ann. of Math. (2), 101:491-498, 1975. MR 51:12535
  • 12. Thomas Jech. Singular cardinal problem: Shelah's theorem on $2\sp {\aleph\sb \omega}$. Bull. London Math. Soc., 24(2):127-139, 1992. MR 93a:03050
  • 13. Kanji Namba. Independence proof of $(\omega ,\,\omega \sb{\alpha })$-distributive law in complete Boolean algebras. Comment. Math. Univ. St. Paul., 19:1-12, 1971. MR 45:6602
  • 14. Kanji Namba. $(\omega \sb{1},\,2)$-distributive law and perfect sets in generalized Baire space. Comment. Math. Univ. St. Paul., 20:107-126, 1971/72. MR 45:8593
  • 15. Matatyahu Rubin and Saharon Shelah. Combinatorial problems on trees: partitions, $\delta$-systems and large free subtrees. Ann. Pure Appl. Logic, 33(1):43-81, 1987. MR 88h:04005
  • 16. Stewart Shapiro (ed.) The limits of logic: Second order logic and the Skolem paradox. The international research library of philosophy. Dartmouth Publishing Company, 1996.
  • 17. Saharon Shelah. Proper forcing. Springer-Verlag, Berlin, 1982. MR 84h:03002
  • 18. Saharon Shelah. Cardinal arithmetic. The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 96e:03001
  • 19. Saharon Shelah. On what I do not understand (and have something to say). I. Fund. Math., 166(1-2):1-82, 2000. Saharon Shelah's anniversary issue. MR 2002a:03091
  • 20. Jack Silver. On the singular cardinals problem. In Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, pp. 265-268. Canad. Math. Congress, Montreal, Que., 1975. MR 55:2576
  • 21. Thoralf Skolem.
    Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen.
    Skrifter utgit av Videnskabsselskapet i Kristiania, I. Matematisk-naturvidenskabelig klasses, 4. 1920.
  • 22. Stevo Todorcevic. Conjectures of Rado and Chang and cardinal arithmetic. In Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), pp. 385-398. Kluwer Acad. Publ., Dordrecht, 1993. MR 95h:03113
  • 23. Boban Velickovic. Forcing axioms and stationary sets. Adv. Math., 94(2):256-284, 1992. MR 93k:03045
  • 24. W. Hugh Woodin. The axiom of determinacy, forcing axioms, and the nonstationary ideal. Walter de Gruyter & Co., Berlin, 1999. MR 2001e:03001

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03C55

Retrieve articles in all journals with MSC (2000): 03C55

Additional Information

Matthew Foreman
Affiliation: Department of Mathematics, University of California, Irvine, California 92697

Stevo Todorcevic
Affiliation: CNRS, Université Paris VII, 2 Place Jussieu, 75251 Paris Cedex 05, Paris, France

Received by editor(s): June 26, 2002
Published electronically: December 16, 2004
Additional Notes: The first author was partially supported by NSF grant DMS-9803126 and the Equipe d’Analyse, Université Paris VI
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society