AMS Home Publications Membership Meetings & Conferences News & Public Outreach Notices of the AMS The Profession Programs Government Relations Education Giving to the AMS About the AMS
MathSciNet® Member Directory Bookstore Journals Employment Services Giving to the AMS
Bookstore MathSciNet® Meetings Journals Membership Employment Services Giving to the AMS About the AMS
The AMS website will be down on Saturday December 11th from 8:30 am to approximately 11:30 am for maintenance.
ISSN 1088-6850(online) ISSN 0002-9947(print)
Journals Home Search My Subscriptions Subscribe
Your device is paired with
for another days.
Previous issue | This issue | Most recent issue | All issues (1900–Present) | Next issue | Previous article | Articles in press | Recently published articles | Next article
$L_{\infty }{}_{\lambda }$-equivalence, isomorphism and potential isomorphism
Authors: Mark Nadel and Jonathan Stavi
Journal: Trans. Amer. Math. Soc. 236 (1978), 51-74
MSC: Primary 02H10; Secondary 02K05, 02H13
DOI: https://doi.org/10.1090/S0002-9947-1978-0462942-3
MathSciNet review: 0462942
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: It is well known that two structures are ${L_{\infty \omega }}$-equivalent iff they are potentially isomorphic [that is, isomorphic in some (Cohen) extension of the universe]. We prove that no characterization of ${L_{\infty \lambda }}$-equivalence along these lines is possible (at least for successor cardinals $\lambda$) and the potential-isomorphism relation that naturally comes to mind in connection with ${L_{\infty \lambda }}$ is often not even transitive and never characterizes ${ \equiv _{\infty \lambda }}$ for $\lambda > \omega$. A major part of the work is the construction of ${\kappa ^ + }$-like linear orderings (also Boolean algebras) A, B such that ${N_{{\kappa ^ + }}}({\mathbf {A}},{\mathbf {B}})$, where ${N_\lambda }({\mathbf {A}},{\mathbf {B}})$ means: A and B are nonisomorphic ${L_{\infty \lambda }}$-equivalent structures of cardinality $\lambda$.
References [Enhancements On Off] (What's this?)
K. Jon Barwise, Absolute logics and $L_{\infty \omega }$, Ann. Math. Logic 4 (1972), 309–340. MR 337483, DOI https://doi.org/10.1016/0003-4843%2872%2990002-2
Jon Barwise, Back and forth through infinitary logic, Studies in model theory, Math. Assoc. Amer., Buffalo, N.Y., 1973, pp. 5–34. MAA Studies in Math., Vol. 8. MR 0342370
K. Jon Barwise, Axioms for abstract model theory, Ann. Math. Logic 7 (1974), 221–265. MR 376337, DOI https://doi.org/10.1016/0003-4843%2874%2990016-3
M. Benda, Reduced products and nonstandard logics, J. Symbolic Logic 34 (1969), 424–436. MR 250860, DOI https://doi.org/10.2307/2270907
J. E. Baumgartner, L. A. Harrington, and E. M. Kleinberg, Adding a closed unbounded set, J. Symbolic Logic 41 (1976), no. 2, 481–482. MR 434818, DOI https://doi.org/10.2307/2272248
Jon Barwise (ed.), The syntax and semantics of infinitary languages, Lecture Notes in Mathematics, No. 72, Springer-Verlag, Berlin-New York, 1968. MR 0234827
C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973. J. Conway, Ph.D. Thesis, Cambridge, England, 196-.
Paul C. Eklof, On the existence of $\kappa $-free abelian groups, Proc. Amer. Math. Soc. 47 (1975), 65–72. MR 379694, DOI https://doi.org/10.1090/S0002-9939-1975-0379694-0
J. Gregory and L. Harrington (unpublished).
Haim Gaifman and E. P. Specker, Isomorphism types of trees, Proc. Amer. Math. Soc. 15 (1964), 1–7. MR 168484, DOI https://doi.org/10.1090/S0002-9939-1964-0168484-2
Jaakko Hintikka and Veikko Rantala, A new approach to infinitary languages, Ann. Math. Logic 10 (1976), no. 1, 95–115. MR 439576, DOI https://doi.org/10.1016/0003-4843%2876%2990026-7
John E. Hutchinson, Model theory via set theory, Israel J. Math. 24 (1976), no. 3-4, 286–304. MR 437336, DOI https://doi.org/10.1007/BF02834760
Thomas J. Jech, Lectures in set theory, with particular emphasis on the method of forcing, Lecture Notes in Mathematics, Vol. 217, Springer-Verlag, Berlin-New York, 1971. MR 0321738
---, The axiom of choice, North-Holland, Amsterdam, 1973.
R. Björn Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR 309729, DOI https://doi.org/10.1016/0003-4843%2872%2990001-0
A Mekler, Ph.D. Thesis, Stanford Univ., Calif., 1976.
J. A. Makowsky, Saharon Shelah, and Jonathan Stavi, $\Delta $-logics and generalized quantifiers, Ann. Math. Logic 10 (1976), no. 2, 155–192. MR 457146, DOI https://doi.org/10.1016/0003-4843%2876%2990021-8
M. Nadel, Model theory in admissible sets, Ph.D. Thesis, Univ. of Wisconsin, 1971.
Mark Nadel, Scott sentences and admissible sets, Ann. Math. Logic 7 (1974), 267–294. MR 384471, DOI https://doi.org/10.1016/0003-4843%2874%2990017-5
M. Nadel and J. Stavi, ${L_{\infty \lambda }}$-equivalence, isomorphism and potential isomorphism of structures, Notices Amer. Math. Soc. 22 (1975), p. A-644. Abstract #75T-E59. J. B. Paris, Solution to a problem of Max Dickman (to appear).
Gerald E. Sacks, Saturated model theory, W. A. Benjamin, Inc., Reading, Mass., 1972. Mathematics Lecture Note Series. MR 0398817
J. R. Shoenfield, Unramified forcing, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 357–381. MR 0280359
J. Stavi, Superstationary sets and their applications (unpublished). ---, On ${L_{\infty \lambda }}$-equivalence of Boolean algebra rings and groups, Notices Amer. Math. Soc. 22 (1975), A-714. Abstract 75T-E77. W. W. Tait, Equivalence in ${L_{\infty \lambda }}$ and isomorphism (to appear).
K. J. Barwise, Absolute logics and ${L_{\infty \omega }}$, Ann. Math. Logic 4 (1972), 309-340. MR 49 #2252. ---, Back and forth through infinitary logic, Studies in Model Theory (M. Morley, ed.), MAA Studies in Math., vol. 8, Math. Assoc. Amer., Buffalo, New York, 1973, pp. 5-34. MR 49 #7116. ---, Axioms for abstract model theory, Ann. Math. Logic 7 (1974), 221-265. M. Benda, Reduced products and nonstandard logics, J. Symbolic Logic 34 (1969), 424-436. MR 40 #4092. J. Baumgartner, L. Harrington and E. M. Kleinberg, Adding a closed unbounded set, J. Symbolic Logic 41 (1976), 481-482. C. C. Chang, Some remarks on the model theory of infinitary languages, The Syntax and Semantics of Infinitary Languages (J. Barwise, ed.), Lecture Notes in Math., vol. 72, Springer-Verlag, Berlin and New York, 1968, pp. 36-63. C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973. J. Conway, Ph.D. Thesis, Cambridge, England, 196-. P. Eklof, On the existence of K-free abelian groups, Proc. Amer. Math. Soc. 47 (1975), 65-72. J. Gregory and L. Harrington (unpublished). H. Gaifman and E. P. Specker, Isomorphism types of trees, Proc. Amer. Math. Soc. 15 (1964), 1-7. MR 29 #5746. J. Hintikka and V. Rantala, A new approach to infinitary languages, Ann. Math. Logic 10 (1976), 95-115. J. Hutchinson, Model theory via set theory (to appear). T. J. Jech, Lectures in set theory, with particular emphasis on the method of forcing, Lecture Notes in Math., vol. 217, Springer-Verlag, Berlin and New York, 1971. MR 48 # 105. ---, The axiom of choice, North-Holland, Amsterdam, 1973. R. B. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229-308. MR 46 #8834. H. J. Keisler, Formulas with linearly ordered quantifiers, The Syntax and Semantics of Infinitary Languages (J. Barwise, ed.), Lecture Notes in Math., vol. 72, Springer-Verlag, Berlin and New York, 1968, pp. 96-130. A Mekler, Ph.D. Thesis, Stanford Univ., Calif., 1976. J. A Makowski, S. Shelah and J. Stavi, $\Delta$-logics and generalized quantifiers, Ann. Math. Logic 10 (1976), 155-192. M. Nadel, Model theory in admissible sets, Ph.D. Thesis, Univ. of Wisconsin, 1971. ---, Scott sentences and admissible sets, Ann. Math. Logic 7 (1974), 264-294. M. Nadel and J. Stavi, ${L_{\infty \lambda }}$-equivalence, isomorphism and potential isomorphism of structures, Notices Amer. Math. Soc. 22 (1975), p. A-644. Abstract #75T-E59. J. B. Paris, Solution to a problem of Max Dickman (to appear). G. E. Sacks, Saturated model theory, Benjamin, New York, 1972. J. R. Shoenfield, Unramified forcing, Axiomatic Set Theory (Proc. Sympos. Pure Math., vol. 13, Part I), Amer. Math. Soc., Providence, R.I., 1971, pp. 357-381. MR 43 #6079. J. Stavi, Superstationary sets and their applications (unpublished). ---, On ${L_{\infty \lambda }}$-equivalence of Boolean algebra rings and groups, Notices Amer. Math. Soc. 22 (1975), A-714. Abstract 75T-E77. W. W. Tait, Equivalence in ${L_{\infty \lambda }}$ and isomorphism (to appear).
Retrieve articles in Transactions of the American Mathematical Society with MSC: 02H10, 02K05, 02H13
Retrieve articles in all journals with MSC: 02H10, 02K05, 02H13
Article copyright: © Copyright 1978 American Mathematical Society
Join the AMS
AMS Conferences
News & Public Outreach
Math in the Media
Mathematical Imagery
Mathematical Moments
Data on the Profession
Fellows of the AMS
Mathematics Research Communities
AMS Fellowships
Collaborations and position statements
Appropriations Process Primer
Congressional briefings and exhibitions
About the AMS
Jobs at AMS
Notices of the AMS · Bulletin of the AMS
American Mathematical Society · 201 Charles Street Providence, Rhode Island 02904-2213 · 401-455-4000 or 800-321-4267
AMS, American Mathematical Society, the tri-colored AMS logo, and Advancing research, Creating connections, are trademarks and services marks of the American Mathematical Society and registered in the U.S. Patent and Trademark Office.
© Copyright , American Mathematical Society · Privacy Statement · Terms of Use · Accessibility