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On equivalence relations second order definable over H(κ)

100%

Shelah S., Vaisanen P.

Fundamenta Mathematicae

2002

tom 174

nr 1

1-21

Let κ be an uncountable regular cardinal. Call an equivalence relation on functions from κ into 2 second order definable over H(κ) if there exists a second order sentence ϕ and a parameter P ⊆ H(κ) such that functions f and g from κ into 2 are equivalent iff the structure ⟨ H(κ), ∈, P, f, g ⟩ satisfies ϕ. The possible numbers of equivalence classes of second order definable equivalence relations include all the nonzero cardinals at most κ⁺. Additionally, the possibilities are closed under unions and products of at most κ cardinals. We prove that these are the only restrictions: Assuming that GCH holds and λ is a cardinal with $λ^κ = λ$, there exists a generic extension where all the cardinals are preserved, there are no new subsets of cardinality < κ, $2^κ = λ$, and for all cardinals μ, the number of equivalence classes of some second order definable equivalence relation on functions from κ into 2 is μ iff μ is in Ω, where Ω is any prearranged subset of λ such that 0 ∉ Ω, Ω contains all the nonzero cardinals ≤ κ⁺, and Ω is closed under unions and products of at most κ cardinals.

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

100%

Shelah S., Vaisanen P.

Fundamenta Mathematicae

2002

tom 174

nr 2

97-126

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 Σ¹₁-definable over $V_κ$. By [SV] it is possible to have a generic extension where the possible numbers of equivalence classes of Σ¹₁-equivalence relations are in a prearranged set. Together these results settle the problem of the possible values of No(M) for models of weakly compact cardinality.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Shelah S.

2 Vaisanen P.

2 2002

Wyniki wyszukiwania

On equivalence relations second order definable over H(κ)

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