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## The number of $L_{\infty\kappa}$-equivalent nonisomorphic models for $\kappa$ weakly compact

### Tom 174 / 2002

Fundamenta Mathematicae 174 (2002), 97-126 MSC: Primary 03C55; Secondary 03C75. DOI: 10.4064/fm174-2-1

#### Streszczenie

For a cardinal $\kappa$ and a model $M$ of cardinality $\kappa$ let ${\rm No}(M)$ denote the number of nonisomorphic models of cardinality $\kappa$ which are $L_{\infty,\kappa}$-equivalent to $M$. We prove that for $\kappa$ a weakly compact cardinal, the question of the possible values of ${\rm No}(M)$ for models $M$ of cardinality $\kappa$ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are $\Sigma^1_1$-definable over $V_\kappa$. By \cite {ShVa719} it is possible to have a generic extension where the possible numbers of equivalence classes of $\Sigma^1_1$-equivalence relations are in a prearranged set. Together these results settle the problem of the possible values of ${\rm No}(M)$ for models of weakly compact cardinality.

#### Autorzy

• Saharon ShelahInstitute of Mathematics
The Hebrew University
Jerusalem, Israel
and
Department of Mathematics
Rutgers University
New Brunswick, NJ 08903, U.S.A.
e-mail
• Pauli VaisanenDepartment of Mathematics
P.O. Box 4
00014 University of Helsinki, Finland
e-mail

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