### The Ehrenfeucht-Fra{\"\i}ss{\'e}-game of length $\omega_1$

by Mekler and Shelah and Vaananen. [MShV:416]

Transactions American Math Soc, 1993

Let (A) and (B) be two first order structures of
the same vocabulary. We shall consider the Ehrenfeucht-Fra{i}sse-game
of
length omega_1 of A and B which we denote by
G_{omega_1}(A, B) . This game is like the ordinary
Ehrenfeucht-Fra{i}sse-game of L_{omega omega} except that
there are omega_1 moves. It is clear that
G_{omega_1}(A, B) is determined if A and B
are of cardinality <= aleph_1 . We prove the following results:
Theorem A: If V=L, then there are models A and B of cardinality
aleph_2 such that the game G_{omega_1}(A, B) is
non-determined.
Theorem B: If it is consistent that there is a measurable cardinal,
then it is consistent that G_{omega_1}(A, B) is
determined for all A and B of cardinality <= aleph_2 .
Theorem C: For any kappa >= aleph_3 there are A and B
of cardinality kappa such that the game G_{omega_1}(A, B)
is non-determined.

Back to the list of publications