Number of models inL∞,ω1theories. II

Palyutin, E. A.

doi:10.1007/bf01669282

Cited by 3 publications

(5 citation statements)

References 3 publications

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“…( 4) (F= L). //λ > K o to rβgwter mtf weαfc/y compact then SP£ λ = {l,2 λ } (67 Palyutin [4] /or λ = «! by Shelah [6] generally), (5) //λ > K o to weakly compact then every χ,2 < χ < λ, belong to SP^0^0 (by Shelah [7]).…”

Section: Theoremmentioning

confidence: 99%

“…(a) g is an isomorphism from Ml [JPf 4 onto N\ {JPJ* (for some /) j<i j<i which is a winning position for player II in the game from 2'.l(b). (j8) the set {<<?,£> : <g(«),ξ> ^/(<tf,£>), e.g., one is defined the other not} is a small subset of M*.…”

Section: Factmentioning

confidence: 99%

“…aGSAβeSΛa<β=>β> Max U a . So if a < β are in 5, by the equation [4], h ai β(Wβ) has elements of the form a^β or αj >7 : (γ</3) only.…”

Section: [2] Asj-afj=w I -H U (Wj)mentioning

confidence: 99%

“…(*) For every α, M\ \J Pf 4 is isomorphic to M a \ \J Gf and let the isomorphism be g~ι and prove M = M b for some b G Gr(2l); by 3.12 (applied to the various 21 ί δ), b will be as required.…”

Section: Factmentioning

confidence: 99%

“…(3) Let RK£ K ={M:MeKl K9 Σ{\R M \:R GL(M)} <λ} RSP£ K = {no(M):M (Ξ RK^J. (4) We always assume that λ, μ, K are >K 0 , K < λ and that μ > c/ K or K is a successor (otherwise MeK£ κ &ME U#£A So w.l.o.g. every MeKμ tK is an L^-model with a fixed L^> κ , which has for a closed unbounded set of OL< K exactly μ α-place predicates when K is a limit cardinal, and μ κ~ place relations when /c = (κ~) + .…”

mentioning

confidence: 99%

See 4 more Smart Citations

On the possible number ${\rm no}(M)=$ the number of nonisomorphic models $L_{\infty,\lambda}$-equivalent to $M$ of power $\lambda$, for $\lambda$ singular.

Shelah¹

1985

Notre Dame J. Formal Logic

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Section: Theoremmentioning

confidence: 99%

Section: Factmentioning

confidence: 99%

“…aGSAβeSΛa<β=>β> Max U a . So if a < β are in 5, by the equation [4], h ai β(Wβ) has elements of the form a^β or αj >7 : (γ</3) only.…”

Section: [2] Asj-afj=w I -H U (Wj)mentioning

confidence: 99%

Section: Factmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

On the possible number ${\rm no}(M)=$ the number of nonisomorphic models $L_{\infty,\lambda}$-equivalent to $M$ of power $\lambda$, for $\lambda$ singular.

Shelah¹

1985

Notre Dame J. Formal Logic

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On the no(M) for M of singular power

Shelah¹

1986

Lecture Notes in Mathematics

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

Shelah

Väisänen

2000

J. symb. log.

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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

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Number of models inL∞,ω1theories. II

Cited by 3 publications

References 3 publications

On the possible number ${\rm no}(M)=$ the number of nonisomorphic models $L_{\infty,\lambda}$-equivalent to $M$ of power $\lambda$, for $\lambda$ singular.

On the possible number ${\rm no}(M)=$ the number of nonisomorphic models $L_{\infty,\lambda}$-equivalent to $M$ of power $\lambda$, for $\lambda$ singular.

On the no(M) for M of singular power

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

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Number of models inL∞,ω1theories. II

Cited by 3 publications

References 3 publications

On the possible number ${\rm no}(M)=$ the number of nonisomorphic models $L_{\infty,\lambda}$-equivalent to $M$ of power $\lambda$, for $\lambda$ singular.

On the possible number ${\rm no}(M)=$ the number of nonisomorphic models $L_{\infty,\lambda}$-equivalent to $M$ of power $\lambda$, for $\lambda$ singular.

On the no(M) for M of singular power

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

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