$L_{\infty }{}_{\lambda }$-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 $\lambda$.

- K. Jon Barwise,
*Absolute logics and $L_{\infty \omega }$*, Ann. Math. Logic**4**(1972), 309β340. 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. 5β34. MAA Studies in Math., Vol. 8. MR**0342370** - K. Jon Barwise,
*Axioms for abstract model theory*, Ann. Math. Logic**7**(1974), 221β265. 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), 424β436. 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, 481β482. 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, - Paul C. Eklof,
*On the existence of $\kappa $-free abelian groups*, Proc. Amer. Math. Soc.**47**(1975), 65β72. MR**379694**, DOI https://doi.org/10.1090/S0002-9939-1975-0379694-0
J. Gregory and L. Harrington (unpublished).
- Haim Gaifman and E. P. Specker,
*Isomorphism types of trees*, Proc. Amer. Math. Soc.**15**(1964), 1β7. MR**168484**, DOI https://doi.org/10.1090/S0002-9939-1964-0168484-2 - Jaakko Hintikka and Veikko Rantala,
*A new approach to infinitary languages*, Ann. Math. Logic**10**(1976), no. 1, 95β115. MR**439576**, DOI https://doi.org/10.1016/0003-4843%2876%2990026-7 - John E. Hutchinson,
*Model theory via set theory*, Israel J. Math.**24**(1976), no. 3-4, 286β304. MR**437336**, DOI https://doi.org/10.1007/BF02834760 - Thomas J. Jech,
*Lectures in set theory, with particular emphasis on the method of forcing*, Lecture Notes in Mathematics, Vol. 217, Springer-Verlag, Berlin-New York, 1971. MR**0321738**
---, - R. BjΓΆrn Jensen,
*The fine structure of the constructible hierarchy*, Ann. Math. Logic**4**(1972), 229β308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR**309729**, DOI https://doi.org/10.1016/0003-4843%2872%2990001-0 - Jon Barwise (ed.),
*The syntax and semantics of infinitary languages*, Lecture Notes in Mathematics, No. 72, Springer-Verlag, Berlin-New York, 1968. MR**0234827**
A Mekler, Ph.D. Thesis, Stanford Univ., Calif., 1976.
- J. A. Makowsky, Saharon Shelah, and Jonathan Stavi,
*$\Delta $-logics and generalized quantifiers*, Ann. Math. Logic**10**(1976), no. 2, 155β192. MR**457146**, DOI https://doi.org/10.1016/0003-4843%2876%2990021-8
M. Nadel, - Mark Nadel,
*Scott sentences and admissible sets*, Ann. Math. Logic**7**(1974), 267β294. MR**384471**, DOI https://doi.org/10.1016/0003-4843%2874%2990017-5
M. Nadel and J. Stavi, ${L_{\infty \lambda }}$- - 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. 357β381. MR**0280359**
J. Stavi,

*Model theory*, North-Holland, Amsterdam, 1973. J. Conway, Ph.D. Thesis, Cambridge, England, 196-.

*The axiom of choice*, North-Holland, Amsterdam, 1973.

*Model theory in admissible sets*, Ph.D. Thesis, Univ. of Wisconsin, 1971.

*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).

*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

Additional Information

Article copyright:
© Copyright 1978
American Mathematical Society