On the Number of Automorphisms of Uncountable Models

Let σ(A) denote the number of automorphisms of a model A of power ω1 . We derive a necessary and sufficient condition in terms of trees for the existence of an A with ω1 < σ(A) < 2 ω1 . We study the sufficiency of some conditions for σ(A) = 21 . These conditions are analogous to conditions studied by D.Kueker in connection with countable models. The starting point of this paper was an attempt to generalize some results of D.Kueker [8] to models of power ω1 . For example, Kueker shows that for countable A the number σ(A) of automorphisms of A is either ≤ ω or 2 . In Corollary 13 we prove the analogue of this result under the set-theoretical assumption I(ω) : if I(ω) holds and the cardinality of A is ω1 , then σ(A) ≤ ω1 or σ(A) = 2 ω1 . In Theorem 16 we show that the consistency strength of this statement + 21 > ω2 is that of an inaccessible cardinal. We use ||A|| to denote the universe of a model A and |A| to denote the cardinality of ||A|| . Kueker proves also that if |A| ≤ ω , |B| > ω and A ≡ B (in L∞ω ), then σ(A) = 2 ω . Theorem 1 below generalizes this to power ω1 . If A and B are countable, A 6= B and A ≺ B (in L∞ω ), then we know that σ(A) = 2 . Theorem 7 shows that the natural analogue of this result fails for models of power ω1 . Theorem 14 links the existence of a model A such that |A| = ω1 , ω1 < σ(A) < 2 ω1 , to the existence of a tree T which is of power ω1 , of height ω1 and has σ(A) uncountable branches. We use A ≡ω1 B to denote that ∃ has a winning strategy in the EhrenfeuchtFräıssé game G(A,B) of length ω1 between A and B . During this game two players ∃ and ∀ extend a countable partial isomorphism π between A and B . At the start of the game π is empty. Player ∀ begins the game by choosing an element a in either A or B . Then ∃ has to pick an element b in either A or B so that a and b are in different models. Suppose that a ∈ A . If the relation π ∪ {(a, b)} is not a partial isomorphism, then ∃ loses immediately, else the game continues in the same manner and the new value of π is the mapping π ∪{(a, b)} . The case a ∈ B is treated similarly, but we consider the relation π ∪ {(b, a)} . The length of our game is ω1 moves. Player ∃ wins, if he can move ω1 times without losing. The only difference between this game and the ordinary game characterizing partial * The first author would like to thank the United States–Israel Binational Science Foundation for support of this research (Publication # 377). The second and third author were supported by Academy of Finland grant 1011040.