2 9 N ov 1 99 9 The number of L ∞ κ - equivalent non - isomorphic models for κ weakly compact
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چکیده
For a cardinal κ and a model M of cardinality κ let No(M) denote the number of non-isomorphic models of cardinality κ which are L∞κ-equivalent to M. In [She82] Shelah established that when κ is a weakly compact cardinal and μ ≤ κ is a nonzero cardinal, there exists a model M of cardinality κ with No(M) = μ. We prove here that if κ is a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ 1 -definable over Vκ. In [SVa] we proved that, consistent wise, the possible numbers of equivalence classes of Σ 1 -equivalence relations can be completely controlled under the singular cardinal hypothesis. These results settle the problem of the possible values of No(M) for models of weakly compact cardinality, provided that the singular cardinal hypothesis holds. 1
منابع مشابه
The number of L∞κ-equivalent non-isomorphic models for κ weakly compact
For a cardinal κ and a model M of cardinality κ let No(M) denote the number of non-isomorphic models of cardinality κ which are L∞,κequivalent to M . We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ1-definable ...
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