The number of $L_{\infty \kappa }$-equivalent nonisomorphic models for κ weakly compact

Saharon Shelah, and Pauli Vaisanen. "The number of $L_{∞κ}$-equivalent nonisomorphic models for κ weakly compact." Fundamenta Mathematicae 174.2 (2002): 97-126. <http://eudml.org/doc/283047>.

@article{SaharonShelah2002,
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 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.},
author = {Saharon Shelah, Pauli Vaisanen},
journal = {Fundamenta Mathematicae},
keywords = {number of models; infinitary logic},
language = {eng},
number = {2},
pages = {97-126},
title = {The number of $L_\{∞κ\}$-equivalent nonisomorphic models for κ weakly compact},
url = {http://eudml.org/doc/283047},
volume = {174},
year = {2002},
}

TY - JOUR
AU - Saharon Shelah
AU - Pauli Vaisanen
TI - The number of $L_{∞κ}$-equivalent nonisomorphic models for κ weakly compact
JO - Fundamenta Mathematicae
PY - 2002
VL - 174
IS - 2
SP - 97
EP - 126
AB - 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.
LA - eng
KW - number of models; infinitary logic
UR - http://eudml.org/doc/283047
ER -