On equivalence relations second order definable over H(κ)

• Autorzy: Saharon Shelah , Pauli Vaisanen
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 03C75: Other infinitary logic
• 03E35: Consistency and independence results
• 03C55: Set-theoretic model theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2002
• Tom: 174
• Numer: 1
• Strony: 1-21

Daty

wydano

2002

Twórcy

autor

Saharon Shelah

• Institute of Mathematics, The Hebrew University, Jerusalem, Israel
• Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A.

autor

Pauli Vaisanen

• Department of Mathematics, P.O. Box 4, 00014 University of Helsinki, Finland

Identyfikatory

DOI

10.4064/fm174-1-1