More on the Ehrenfeucht–Fraisse game of length $\omega _1$

Tapani Hyttinen¹ ; Saharon Shelah² ; Jouko Vaananen¹

¹ Department of Mathematics P.O. Box 4 (Yliopistonkatu 5) 00014 University of Helsinki, Finland
² Einstein Institute of Mathematics The Hebrew University of Jerusalem Jerusalem 91904, Israel and Deparment of Mathematics Rutgers University New Brunswick, NJ 08903, U.S.A.

Fundamenta Mathematicae, Tome 175 (2002) no. 1, pp. 79-96.

Résumé

By results of [9] there are models ${\frak A}$ and ${\frak B}$ for which the Ehrenfeucht–Fraïssé game of length $\omega _1$, ${\rm EFG}_{\omega _1}({\frak A},{\frak B})$, is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality $\le \aleph _2$. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement “CH and ${\rm EFG}_{\omega _1}({\frak A},{\frak B})$ is determined for all models ${\frak A}$ and ${\frak B}$ of cardinality $\aleph _2$” is that of a weakly compact cardinal. On the other hand, we show that if $2^{\aleph _0}2^{\aleph _{3}}$, $T$ is a countable complete first order theory, and one of (i) $T$ is unstable, (ii) $T$ is superstable with DOP or OTOP, (iii) $T$ is stable and unsuperstable and $2^{\aleph _0}\le \aleph _{3}$, holds, then there are ${\cal A},{\cal B}\mathrel |\mathrel {\mkern -3mu}=T$ of power $\aleph _{3}$ such that ${\rm EFG}_{\omega _{1}}({\cal A},{\cal B})$ is non-determined.

Détail

DOI : 10.4064/fm175-1-5

Keywords: results there models frak frak which ehrenfeucht fra game length omega efg omega frak frak non determined consistent relative consistency measurable cardinal models have cardinality aleph improve work ways firstly prove consistency strength statement efg omega frak frak determined models frak frak cardinality aleph weakly compact cardinal other aleph aleph countable complete first order theory unstable superstable dop otop iii stable unsuperstable aleph aleph holds there cal cal mathrel mathrel mkern power aleph efg omega cal cal non determined