Almost free groups and Ehrenfeucht-Fra{\"\i}ss\'e games for successors of singular cardinals

by Shelah and Vaisanen. [ShVs:787]
Annals Pure and Applied Logic, 2002
We strengthen non-structure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fra{i}sse games between a fixed group of cardinality lambda and a free Abelian group. A group is called epsilon-game-free if the isomorphism player has a winning strategy in the game (of the described form) of length epsilon in lambda . We prove for a large set of successor cardinals lambda = mu^+ existence of nonfree (mu . omega_1)-game-free groups of cardinality lambda . We concentrate on successors of singular cardinals.

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