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Transactions of the American Mathematical Society

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$ L\sb{\infty }{}\sb{\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
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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 $.


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  • [Ba1] K. J. Barwise, Absolute logics and $ {L_{\infty \omega }}$, Ann. Math. Logic 4 (1972), 309-340. MR 49 #2252. MR 0337483 (49:2252)
  • [Ba2] -, 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. MR 0342370 (49:7116)
  • [Ba3] -, Axioms for abstract model theory, Ann. Math. Logic 7 (1974), 221-265. MR 0376337 (51:12513)
  • [Be] M. Benda, Reduced products and nonstandard logics, J. Symbolic Logic 34 (1969), 424-436. MR 40 #4092. MR 0250860 (40:4092)
  • [BHK] J. Baumgartner, L. Harrington and E. M. Kleinberg, Adding a closed unbounded set, J. Symbolic Logic 41 (1976), 481-482. MR 0434818 (55:7782)
  • [Ch] 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. MR 0234827 (38:3141)
  • [CK] C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.
  • [Co] J. Conway, Ph.D. Thesis, Cambridge, England, 196-.
  • [Ek] P. Eklof, On the existence of K-free abelian groups, Proc. Amer. Math. Soc. 47 (1975), 65-72. MR 0379694 (52:599)
  • [GH] J. Gregory and L. Harrington (unpublished).
  • [GS] H. Gaifman and E. P. Specker, Isomorphism types of trees, Proc. Amer. Math. Soc. 15 (1964), 1-7. MR 29 #5746. MR 0168484 (29:5746)
  • [HR] J. Hintikka and V. Rantala, A new approach to infinitary languages, Ann. Math. Logic 10 (1976), 95-115. MR 0439576 (55:12462)
  • [Hu] J. Hutchinson, Model theory via set theory (to appear). MR 0437336 (55:10268)
  • [Jel] 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. MR 0321738 (48:105)
  • [Je2] -, The axiom of choice, North-Holland, Amsterdam, 1973.
  • [Jn] R. B. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229-308. MR 46 #8834. MR 0309729 (46:8834)
  • [Ke] 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. MR 0234827 (38:3141)
  • [Mk] A Mekler, Ph.D. Thesis, Stanford Univ., Calif., 1976.
  • [MSS] J. A Makowski, S. Shelah and J. Stavi, $ \Delta$-logics and generalized quantifiers, Ann. Math. Logic 10 (1976), 155-192. MR 0457146 (56:15362)
  • [Na1] M. Nadel, Model theory in admissible sets, Ph.D. Thesis, Univ. of Wisconsin, 1971.
  • [Na2] -, Scott sentences and admissible sets, Ann. Math. Logic 7 (1974), 264-294. MR 0384471 (52:5348)
  • [NS] 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.
  • [Pa] J. B. Paris, Solution to a problem of Max Dickman (to appear).
  • [Sa] G. E. Sacks, Saturated model theory, Benjamin, New York, 1972. MR 0398817 (53:2668)
  • [Sh] 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. MR 0280359 (43:6079)
  • [St1] J. Stavi, Superstationary sets and their applications (unpublished).
  • [St2] -, On $ {L_{\infty \lambda }}$-equivalence of Boolean algebra rings and groups, Notices Amer. Math. Soc. 22 (1975), A-714. Abstract 75T-E77.
  • [Ta] W. W. Tait, Equivalence in $ {L_{\infty \lambda }}$ and isomorphism (to appear).

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DOI: https://doi.org/10.1090/S0002-9947-1978-0462942-3
Article copyright: © Copyright 1978 American Mathematical Society

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