Number of models inL∞,ω1-theories

2002

Fund. Math.

Abstract. For a cardinal κ and a model M of cardinality κ let No(M ) denote the number of nonisomorphic models of cardinality κ which are L ∞,κ -equivalent 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 Σ . This result extends to all structures of singular cardinality λ provided that λ is of countable cofinality [Cha68]. The case where M is of singular cardinality λ with uncountable cofinality κ was first treated in [She85] and later on in [She86]. In these papers Shelah showed that if κ > ℵ 0 , θ κ < λ for every θ < λ, and 0 < µ < λ or µ = λ κ , then No(M) = µ for some model M of cardinality λ. In [SV00] of the present authors the singular case is revisited, and in particular, it is established, under the same assumptions as above, that the values µ with λ ≤ µ < λ κ are possible for No(M) with M of cardinality λ.

“…For κ = ℵ 1 this result was first proved in [Pal77a]. The values No(M) ∈ {ℵ 0 , ℵ 1 } for a model of cardinality ℵ 1 are consistent with ZFC + GCH as noted in [She81].…”

supporting

confidence: 57%

The number of L_∞κ-equivalent nonisomorphic models for κ weakly compact

2002

Fund. Math.

“…For κ = ℵ 1 this result was first proved in [Pal77a]. Later Shelah extended the result to all other regular non-weakly compact cardinals in [She81b].…”

Section: Introductionmentioning

confidence: 81%

On the number of ${L}_{\infty \omega _1}$-equivalent non-isomorphic models

Trans. Amer. Math. Soc.

2000

Abstract. We prove that if ZF is consistent then ZFC + GCH is consistent with the following statement: There is for every k < ω a model of cardinality ℵ 1 which is L∞ω 1 -equivalent to exactly k non-isomorphic models of cardinality ℵ 1 . In order to get this result we introduce ladder systems and colourings different from the "standard" counterparts, and prove the following purely combinatorial result: For each prime number p and positive integer m it is consistent with ZFC + GHC that there is a "good" ladder system having exactly p m pairwise nonequivalent colourings.

“…If V = L and κ ≥ ℵ 1 is a regular cardinal which is not weakly compact, No(M) has either the value 1 or 2 κ for all models M having cardinality κ. For κ = ℵ 1 this result was first proved in [Pal77a]. Later Shelah extended the result to all other regular non-weakly compact cardinals in [She81b].…”

Section: Introductionmentioning

confidence: 81%

“…If V = L, A is an uncountable regular cardinal which is not weakly compact, and M is a model of cardinality A, then No(^#) has either the value 1 or 2 X . For A = tt\ this result was first proved in [2]. Later in [5] Shelah extended this result to all other regular non-weakly compact cardinals.…”

mentioning

confidence: 88%

On inverse γ-systems and the number ofL_∞λ-equivalent, non-isomorphic models for λ singular

2000

J. symb. log.

Suppose λ is a singular cardinal of uncountable cofinality κ. For a model M of cardinality λ, let No(M) denote the number of isomorphism types of models N of cardinality λ which are L ∞λ -equivalent to M. In [She85] Shelah considered inverse κ-systems A of abelian groups and their certain kind of quotient limits Gr(A)/Fact(A). In particular Shelah proved in [She85, Fact 3.10] that for every cardinal µ there exists an inverse κ-system A such that A consists of abelian groups having cardinality at most µ κ and card Gr(A)/Fact(A) = µ. Later in [She86, Theorem 3.3] Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that θ κ < λ for every θ < λ): if A is an inverse κ-system of abelian groups having cardinality < λ, then there is a model M such that card(M) = λ and No(M) = card Gr(A)/Fact(A) . The following was an immediate consequence (when θ κ < λ for every θ < λ): for every nonzero µ < λ or µ = λ κ there is a model M µ of cardinality λ with No(M µ ) = µ. In this paper we show: for every nonzero µ ≤ λ κ there is an inverse κ-system A of abelian groups having cardinality < λ such that card Gr(A)/Fact(A) = µ (under the assumptions 2 κ < λ and θ <κ < λ for all θ < λ when µ > λ), with the obvious new consequence concerning the possible value of No. Specifically, the case No(M) = λ is possible when θ κ < λ for every θ < λ. 1