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Almost free groups and long Ehrenfeucht–Fraı̈ssé games

✍ Scribed by Pauli Väisänen

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
357 KB
Volume
123
Category
Article
ISSN
0168-0072

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✦ Synopsis

An Abelian group G is strongly -free i G is L ∞; -equivalent to a free Abelian group i the isomorphism player has a winning strategy in an Ehrenfeucht-Fra ssà e game of length ! between G and a free Abelian group. We study possible longer Ehrenfeucht-Fra ssà e games between a nonfree group and a free Abelian group. A group G is called -game-free if the isomorphism player has a winning strategy in an Ehrenfeucht-Fra ssà e game of length between G and a free Abelian group. We prove in ZFC existence of nonfree -game-free groups for many successors of regular cardinals. We also show that the length of the game obtained is very close to the optimal length provable in ZFC alone. On the other hand, assuming existence of a Mahlo cardinal, we sketch a proof that it is consistent to have a very highly game-free still nonfree group. First we present an introduction to basic constructions and then we introduce some results concerning the standard tools for building more complicated groups, namely transversals and -systems. It follows that all the constructions generalize to algebras in a ÿxed variety satisfying the strong construction principle.

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## Abstract In this paper we prove under some set theoretical assumptions that if __T__ is a countable unstable theory then there is a pair of models of __T__ such that Ehrenfeucht‐Fraïssé games between these models of large variety of lengths are non‐determined.