Claim 3. The first-order theory of graph is interpretable in the monadic .... Claim 4. Suppose a < X and cf a > co. Then G2(a) is a club subset of a of.
THEJOURNAL
OF SYMBOLIC LOGIC Volume 48, Number 2, June 1983
THE MONADIC THEORY
YURI
GUREVICH,
MENACHEM
MAGIDOR
OF021
AND
SAHARON
SHELAH
Abstract. Assume ZFC + "There is a weakly compact cardinal" is consistent. (i) For every S c C), ZFC + "S and the monadic
the other" is consistent; and (ii) ZFC + "The full second-order theory of
Then: theory of o2 are recursive each in
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 interpretedas arbitrarysubsets of M. Speaking about the monadic theory of an ordinal cowe 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 BUchihad 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 derivativeof 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 U0and 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. ?
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1983, Association for Symbolic Logic 0022-4812/83/4802-0016/$02.20
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YURI GUREVICH, MENACHEMMAGIDOR AND SAHARON SHELAH
Assuming ZFC + "There is a weakly compact cardinal" Magidor proved in [Ma] that ZFC + "D(X) = U1 modulo I for every stationary X c UO"is consistent. By (i) the monadic theory of w2 is decidable in Magidor's universe. In [Sh2] Shelah proved that a certain combinatorial principle (the uniformization property for 02) implies the premise of (ii), and that the uniformization is consistent with ZFC + CH. Later he proved that the uniformization property is consistent even with ZFC + GCH, see [St & Ki]. By (ii) the monadic theory of w2 is decidable in the corresponding universes of Shelah. It is different however from the monadic theory of w2 in Magidor's universe. The first undecidability result on monadic theories of ordinals was presented by Magidor in Logic Colloquium 77 (Wroclaw, Poland, 1977). For n > 2 let En say that for every stationary X ' an, consisting of ordinals of cofinality w there is a < (on,of cofinality > w such that a n xis stationary in a. Assuming consistency of w supercompact cardinals Magidor proved that for every S c w - {O, 1} there is a world with {n: En is true} = S. Shelah proved in this direction the following. Assume consistency of w Mahlo cardinals; then for every S c {2n: 1 < n < w} there is-a world with {2n: E2n is true} = S. And for every S c {2n: 1 < n < A} it is consistent with ZFC + GCH that {2n: 1 < n < w and there is a stationary X C (02n such that for every Y c X there is Z ' (2n with {a < (02n: cf(a) > a) and a n Z is stationary in al}= Y modulo nonstationary sets} is equal to S. None of these three results (one of Magidor and two of Shelah) is published. In Part I of this article we prove in detail: THEOREM 1. Assume there is a weakly compact cardinal. Then there is an algorithm n - on such that on is a sentence in the monadic language of order and for every S c w there is a generic extension of the ground world with {n: 0)2 k= Obn}= S. Thus there are continuum many possible monadic theories of 0)2 (in different universes) and for every S c w there is a monadic theory of (X2 (in some world) which is at least as complex as S. The full second-order theory of a set X is the theory of X in the language with variables for elements, variables for arbitrary monadic predicates, variables for arbitrary dyadic predicates, etc. It depends on the cardinality of X only. It belongs more to set theory than to model theory and can be used as a standard of complexity. The monadic theory of 0)2 is easily interpretable in the full second-order theory of )2. Thus the full second-order theory of 0)2 gives an upper bound of complexity of the monadic theory of )2THEOREM 2. Assume there is a weakly compact cardinal. Then there is a generic extension of the ground world where thefull second-ordertheory of (02 is interpretable (thereforerecursive)in the monadictheory of (2. Theorem 2 is proved in Part II. It has actually the same proof as Theorem 1 with only a few alterations. Combining the technique of Part I and that of [Ma] we prove in Part III THEOREM 3. Assume GCH and existence of a weakly compact cardinal. For every S ' 02 there is a generic extension of the ground universe where S and the monadic theory of 0)2 are recursiveeach in the other. If X is weakly compact in a world V then it is weakly compact in the constructible part of V (see [Je])where GCH holds. Hence Theorem 3 gives
389
THE MONADIC THEORY OF (02 COROLLARY 4.
Assume ZFC + "There is a weakly compact cardinal" is consistent. Thenfor every S c w, ZFC + "S and the monadic theory of 02 are recursive each in the others" is consistent. We use the book [Je]and the article [Sho] as sources of notation, terminology and information. PART I. CONTINUUM POSSIBLE MONADIC THEORIES OF
(02
?1. Coding. Here a graph is a model where R is a reflexive symmetric binary relation on X such that for every different x, y in X there is z E X with Rxz not equivalent to Ryz. Claim 1. There is an algorithm n --+ qn such that (pn is a first-ordergraph sentence and for every S c w there is a graph with {n: satisfies (Pn}= S. PROOF. For every n E w - S add to w an auxiliary element n', for every n E S add two auxiliary elements n' and n". Let R be the least reflexive symmetric relation on the resulting set containing pairs (n, n'), (n, n + 1) for n < w and pairs (n, n") for n E S. It is easy to see that x is auxiliary if it is R-connected with at most two elements including itself. Hence 0, 1, ... are definable. (qnsays that n is Rconnected with two auxiliary elements. It remains only to replace all elements by natural numbers. E] Given a graph and assuming existence of a weakly compact cardinal we define in ?2 a forcing notion P and prove in ?6: THEOREM 2. Suppose G is a P-generic filter over the ground world V. Then in V[G] there is a partition of {a < 02: Cfa = w} into stationary sets S,, n < w, such that (i) for every a < 0)2 of cofinality Wi there is a pair (m, n) E R such that Sm U Sn includes a club subset of a, and (ii) if (m, n) E R, AmC Sm, An C Sn and Am,An are stationary in 02 then there are stationarily many ordinalsa < 2 of cofinality al with both Am n a and A. f a stationary in a. Note that clause (i) of Theorem 2 implies D(Sm)n D(S,) = 0 for (m, n) E (w x ao) - R. Claim 3. The first-order theory of graph is interpretable in the monadic theory of (02if there is a partition described in Theorem 2. PROOF.U0, U1, the ideal I and the derivative D are defined in the Introduction. Here are some more definitions in the monadic theory Of 02. Two subsets X, Y of U0 are connected if D(X) n D(Y) is stationary. (Note that X c U0 is connected with itself iff X is stationary.) A stationary X c U0 is an atom if there are no X1, X2 c X and Y c U0 such that X1, X2 are stationary and Y is connected with one of sets X1, X2 but not with the other. An atom X is maximal if X = Y modulo I for every atom Y including X. For every m every stationary X c Sm is an atom. For, suppose X1, X2 are staS: (m, n) E R}, Y2 = Y tionary subsets of Xand Y c U0. Let Y1 = Y1. If Y1 is stationary then some Y n Sn with (m, n) E R is stationary and Y is connected with both X1 and X2. Otherwise the derivative of Y coincides modulo I with the derivative Y2which avoids even the derivative of X, thus Y is connected with neither X1 nor X2.
U{Yn
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YURI GUREVICH, MENACHEMMAGIDOR AND SAHARON SHELAH
If X is an atom then X c S_ modulo I for some m. For, let K = {m: X n Smis stationary}. K # 0 because X is stationary. If different m, n belong to K there is / such that Rlm is not equivalent to Rln. Set X1 = X n Smu, X2= X n SnoY = so to contradict the assumption that Xis an atom. It is easy to see that an atom Xis maximal iffX = Smmodulo Ifor some m. Now interpret variables of the first-order graph language as maximal atoms (equality is equality modulo I) and R as connectedness. El Claim 1, Theorem 2 and Claim 3 imply THEOREM 4. Assume there is a weakly compact cardinal. There is a recursive list 0b0,sb1, . . . of monadic sentences such thatfor every S c Wthere is a generic extension of the ground world with {n: (02 # Obn} = S. ?2. Forcing notion. Suppose X is a weakly compact cardinal and R is a reflexive symmetric binary relation on w. We define a forcing notion P for collapsing X onto (02 and creating stationary subsets Sm of (02 described in ?1. A condition p is a triple (pO, p1, p2) of countable functions. pO gives a partial information about sets Sm. it is composed of pairs (a, m) where cf a = o, m < w; the intended meaning is a E Sm.p1 assigns a pair (m, n) E R to ordinals a < X of cofinality > o; the intended meaning is: So n a is stationary in a iff 1 e {m, n}. To assure our intentions p2 assigns a closed countable subset of a to each a E dom p1 in such a way that p2(a) c {P: pO(a) is equal to m or to n} where (m, n) = p1(a); the intended meaning is: p2(a) is an initial segment of a club subset of a included in Sm U So. A condition p refines a condition q (p < q) if qO c pO, ql c p1 and for every a E dom p1, p2(a) is an end extension of q2(a) and min(p2(a)
-
q2(a)) > sup(acn (dom pO U dom p1)).
The last requirement is used to prove the following Claim 1. (It is convenient for us to treat elements of R as unordered pairs.) Claim 1. P is w1-closed. n < (}, y= U {pl n < 0} PROOF. Given po > p, > * set x =U{p.O: E dom and z(a) be the closure of U{p,2(a): a pl} for a E dom y. Let d = {a E dom y: sup z(a) does not belong to any p"2(a)}. If d # 0 then (x, y, z) 0 P because sup z(a) 0 dom x for a E d. Select a functions = {(sup z(a), ma): a E d, ma e y(a)}; this is possible because a, P E d, a < P imply sup z(a) < a < sup z(p). (x Uf, y, z) belongs to P and refines any p, El If p E P then dom pO U dom p1 will be called the domain of p and denoted dom(p). sup dom(p) will be called the height ofp and denoted h(p). Claim 2. P satisfies the x-chain condition. PROOF. By contradiction suppose that {pa: a < 4r is a set of pairwise incompatible conditions. Define f (a) = sup(a n dom pa). f is regressive on {a: cf a > c)}. By Fodor's Lemma there is a stationary A s X with f(a) = 3 for some 3 and any There are at most A possibilities for pOl, p1Ij, p2ld. Hence a E A. Let A = 13Iw. there is a such that B = {P e A: pai and ppicoincide on 3 for i < 2) is of cardinality x. There is P1e B exceeding h(pa). Then Pa, Pp are compatible, thus we have a contradiction. []
THE MONADIC THEORYOF (02
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COROLLARY 3. For every P-name a there is a function f: K K such thatfor every a < X and everyp forcing a E a there is q such that h(q) < fa, q is compatiblewith p and qforces a re a. PROOF. Consider a maximal antichain C c P such that every element of C either forces a E a or forces a 0 a. Setfa = sup{h(p): p E C}. [D In the rest of this section G is a P-generic ultrafilter over the ground world V, Gi = U{pi: p E G} for i < 1. It is easy to see that dom GO = {a < x: cf a = w} and dom GI = {a < x: cf a > w}. As P is ol-closed, oVG] = co. Thus cf a > o in Viffcf a > woin V[G].For a < Kwith cf a > colet G2(a) = U{p2(a): p E G}. Claim 4. Suppose a < X and cf a > co. Then G2(a) is a club subset of a of
cofinality (01.
If Ee G2(a) then : E p2(a) for some p E G, hence cfp = co. Let c = e p2(a)}, so that G2(a) is the denotation of c in V[G]. For a E dom p and E p): {(O, < a the set forces {p: r E c for some r > 3} is dense, hence G2(a) is unbounded. p 0 Suppose p < Kis a limit point for G2(a). There are r > p and p E G forcing r E c. Then G2(a) n r = p2(a) n r. As p2(a) is closed, p E p2(a) c G2(a). Thus G2(a) is a club subset of a consisting of ordinals of cofinality co,hence cf G2(a) = oi. EL COROLLARY 5. K is ()2 in V[G]. PROOF. In V[G]: K > (o1 because ()V[G] = 0)V0, K ? (2 by Claim 4, K is a cardinal because P satisfies the K-chaincondition. LI Claim 6. Every new club subset of Kin V[G]includes an old club subset of K. Proof is well known and uses only the K-chaincondition. Suppose the empty condition (0, 0, 0) forces "c is a club subset of K". Let C' = {a: the empty condition forces a E c}. It is obvious that C' is closed. We prove that C' is unbounded. For a < K let A(a) be the set of ordinals p < a such that some p forces "p is the least element of c above a". IA(a)l < K because P satisfies the K-chain condition. Letfa = sup A(a). Now given ao let aOn+ = fan, a = SUp{an: n < (0}. The empty condition forces that c meets every interval (ang a?ni], hence it forces a e c, i.e., ace C'. LI PROOF.
?3. Decomposition of forcing. First we recall the notion of quotient forcing. Suppose B is a partial ordering and A is a submodel of B satisfying the following conditions: if p E A and p < q then q E A; if two elements of A are compatible in B then they are compatible in A; and for each q E B there is a unique p E A (the projection of q) such that q < p and p, q are compatible with exactly the same elements of A. Let c = {(q, p): q E B and p is the projection of q}. For every A-generic filter G over the ground world V, cvC (i.e., the denotation of c in V[G]) is equal to {q E B: q is compatible with any p E G}. This denotation is the quotient forcing in the following sense: B is isomorphic to a dense subset of A * c and forcing with B is a composition of forcing with A and subsequent forcing with the denotation of c. For 0 < a < K let P( < a) be the submodel of P comprising conditions of height < a. For p E P let p( < a) = (pOla,p1Ia, p2la), it is the projection of p into P( < a). For every a < p < K with cf a > cowe get the quotient forcing completing forcing with P( < a) to a forcing with P( < p). Two cases are of special interest for us. Let A < Kbe of cofinality > oand P(
