A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals

Dorottya Sziráki ¹   ; Jouko Väänänen ²

¹ Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Reáltanoda u. 13–15 H-1053 Budapest, Hungary and Department of Mathematics and its Applications Central European University Nádor u. 9 H-1051 Budapest, Hungary
² Department of Mathematics and Statistics Gustaf Hällströmin katu 2b FI-00014 University of Helsinki, Finland and Institute for Logic, Language and Computation University of Amsterdam 1090 GE Amsterdam, Netherlands

Fundamenta Mathematicae, Tome 238 (2017) no. 1, pp. 53-78

Résumé

We consider the following dichotomy for $ {\mathbf {\Sigma }^0_2}$ finitary relations $R$ on analytic subsets of the generalized Baire space for $\kappa $: either all $R$-independent sets are of size at most $\kappa $, or there is a $\kappa $-perfect $R$-independent set. This dichotomy is the uncountable version of a result found in [W. Kubiś, Proc. Amer. Math. Soc. 131 (2003), 619–623] and in [S. Shelah, Fund. Math. 159 (1999), 1–50]. We prove that the above statement holds if we assume $\Diamond _\kappa $ and the set-theoretical hypothesis $ {\mathrm {I}^-(\kappa )}$, which is the modification of the hypothesis $ {\mathrm {I}(\kappa )}$ suitable for limit cardinals. When $\kappa $ is inaccessible, or when $R$ is a closed binary relation, the assumption $\Diamond _\kappa $ is not needed. We obtain as a corollary the uncountable version of a result by G. Sági and the first author [Logic J. IGPL 20 (2012), 1064–1082] about the $\kappa $-sized models of a $ {\mathbf {\Sigma }^1_1}({L_{\kappa ^+\kappa }})$-sentence when considered up to isomorphism, or elementary embeddability, by elements of a $K_\kappa $ subset of ${}^\kappa \kappa $. The elementary embeddings can be replaced by a more general notion that also includes embeddings, as well as the maps preserving $L_{\lambda \mu }$ for $\omega \leq \mu \leq \lambda \leq \kappa $ and finite variable fragments of these logics.

Détail

DOI : 10.4064/fm130-9-2016

Keywords: consider following dichotomy mathbf sigma finitary relations analytic subsets generalized baire space kappa either r independent sets size kappa there kappa perfect r independent set dichotomy uncountable version result found kubi proc amer math soc shelah fund math nbsp nbsp prove above statement holds assume diamond kappa set theoretical hypothesis mathrm kappa which modification hypothesis mathrm kappa suitable limit cardinals kappa inaccessible closed binary relation assumption diamond kappa needed obtain corollary uncountable version result nbsp first author logic igpl nbsp nbsp about kappa sized models mathbf sigma kappa kappa sentence considered isomorphism elementary embeddability elements kappa subset nbsp kappa kappa elementary embeddings replaced general notion includes embeddings maps preserving lambda omega leq leq lambda leq kappa finite variable fragments these logics