The Monadic Theory Of021 Yuri Gurevich, Menachem Magidor and Saharon Shelah

Assume ZFC + "There is a weakly compact cardinal" is consistent. Then: (i) For every S c C), ZFC + "S and the monadic theory of o2 are recursive each in the other" is consistent; and (ii) ZFC + "The full second-order theory of w2 is interpretable in the monadic theory of W2" is consistent. Introduction. First we recall the definition of monadic theories. The monadic language corresponding to a first-order language L is obtained from L by adding variables for sets of elements and adding atomic formulas x E Y. The monadic theory of a model M for L is the theory of M in the described monadic language when the set variables are interpreted as arbitrary subsets of M. Speaking about the monadic theory of an ordinal co we mean the monadic theory of . Formal theories of order were studied very extensively. We do not review that study here. Our attention is restricted to the monadic theory of ordinals. The pioneer here was BUchi. He proved decidability of the monadic theory of w, the monadic theory of w1, and the monadic theory of ordinals < 2. See the strongest result in [Bu]. Note that the last of these theories is not the monadic theory of w)2, but the set of monadic statements true in every ordinal < W2. As we will see below BUchi had a good reason to stop at w2. Shelah studied the monadic theory of w2 in [Shl]. We shall use some of his results. Let U, = {a < w2: cf a = Wi} for i < 1, and I be the ideal of nonstationary sets. For X c U0 let D(X) = {a E Ul: a n X is stationary in a}. We call D(X) the derivative of X. It is easy to see that D(X) = D(Y) modulo I if X = Y modulo I, thus D can be considered as a relation on the Boolean algebra PS((w2)/I of subsets of w2 modulo I. (PS(X) denotes here the power set of X and the corresponding Boolean algebra.) Shelah proved: (i) the monadic theory of 02 and the first-order theory of are recursive each in the other; (ii) the monadic theory of )2 is decidable if for every stationary X c U0 and every Y1, Y2 with D(X) = Y1 U Y2 there are disjoint stationary X1, X2 such that X1 U X2= Xand D(Xj) = Y modulo I for i = 1, 2. He noted also that (Baumgartner and Jensen's results imply that) "2 t= (DX # 0 for every X c U0)" is independent in ZFC. Received November 21, 1980; revised August 30, 1981. 'The results were obtained and the paper was written during the Logic Year in the Institute for Advanced Studies of Hebrew University. ? 1983, Association for Symbolic Logic 0022-4812/83/4802-0016/$02.20