On equivalence relations second order definable over H(κ)

Saharon Shelah, and Pauli Vaisanen. "On equivalence relations second order definable over H(κ)." Fundamenta Mathematicae 174.1 (2002): 1-21. <http://eudml.org/doc/282598>.

@article{SaharonShelah2002,
abstract = {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.},
author = {Saharon Shelah, Pauli Vaisanen},
journal = {Fundamenta Mathematicae},
keywords = {second order definable equivalence relations; number of models; infinitary logic},
language = {eng},
number = {1},
pages = {1-21},
title = {On equivalence relations second order definable over H(κ)},
url = {http://eudml.org/doc/282598},
volume = {174},
year = {2002},
}

TY - JOUR
AU - Saharon Shelah
AU - Pauli Vaisanen
TI - On equivalence relations second order definable over H(κ)
JO - Fundamenta Mathematicae
PY - 2002
VL - 174
IS - 1
SP - 1
EP - 21
AB - 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.
LA - eng
KW - second order definable equivalence relations; number of models; infinitary logic
UR - http://eudml.org/doc/282598
ER -