Abstract
We study the possibilities of constructing, in ZFC without any additional assumptions, strongly equivalent non-isomorphic trees of regular power. For example, we show that there are non-isomorphic trees of power !2 and of height ! ! such that for all < !1 ! !, E has a winning strategy in the Ehrenfeucht{Frass e game of length . The main tool is the notion of a club-guessing sequence. In this paper we study the problem of constructing strongly equivalent non-isomorphic models. In the 60's, the problem was studied in connection with the arising interest in innitary languages. The idea was to study the relation between isomorphism and elementary equivalence in innitary log- ics. Well known contributions were made e.g. by C. Karp and M. Morley. These studies were continued in the 70's, e.g. by M. Nadel and J. Stavi.
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