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Transactions of the American Mathematical Society

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

 
 

 

$L_{\infty }{}_{\lambda }$-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
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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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Article copyright: © Copyright 1978 American Mathematical Society