Proc. NatI. Acad. Sci. USA Vol. 85, pp. 6582-6586, September 1988 Mathematics
Projective determinacy (set theory/descriptive set theory/large crdinals) DONALD A. MARTIN AND JOHN R. STEEL Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90024 Communicated by Robert M. Solovay, May 9, 1988 (receivedfor review October 22, 1987)
ABSTRACT It is shown that projective determinacy fol- set theory. Instead success came by way of a different class lows from large cardinal axioms weaker than the assertion that of axioms: determinacy axioms. Let cl be the set of all natu- supercompact cardinals exist. ral numbers. Z1w is thus the set of all infinite sequences of natural numbers. With the natural topology, E'n is homeo- The ZFC (Zermelo-Fraenkel, with the axiom of choice) axi- morphic to the irrationals. Let A C Zw. We associate with A oms for set theory are quite successful: they provide a foun- a game of infinite length, as follows. Players I and II take dation for mathematics, they are apparently free from con- turns choosing natural numbers, thus producing an x E Z'o. I tradiction and paradox, and they seem intuitively to be true wins if and only if x E A. The notions of strategies and win- of the informal iterative concept of set. Nevertheless the ning strategies are defined in the obvious way. The game ZFC axioms have serious limitations. From the work of Go- associated with A is determined (or, briefly, A is deter- del, Cohen, and-after Cohen-many others, it is known mined) if one of the players has a winning strategy. Here are that a large number of fundamental questions of set theory some examples of determinacy axioms: projective determi- itself cannot be answered either positively or negatively on nacy (PD), every projective subset of Z'o is determined; the basis of the ZFC axioms. The most prominent of these DL(R), every subset of Z'o belonging to L(R), the collection questions is that of the continuum hypothesis (CH) of Can- of sets constructible from the reals, is determined. Even the tor, but others are in some ways more concrete: e.g., ques- full axiom of determinacy (AD), asserting that every subset tions of descriptive set theory, which concern not arbitrary of A'os is determined, was introduced by Mycielski and Stein- sets of real numbers as does the CH but only simply defin- haus (see chapter 7 of ref. 2) and has been extensively stud- able sets of real numbers. ied, despite the fact that it contradicts the axiom of choice. Since the standard axioms are not adequate, it seems rea- Unlike large cardinal axioms, determinacy axioms do not sonable to look for additional axioms. Indeed, a whole class seem to enjoy any direct a priori evidence. Their appeal lies of candidates for axiomhood has been found; the so-called in the fact that, beginning in 1962, it has been shown that large cardinal axioms. Though a great number and variety of determinacy axioms are sufficient to settle virtually every such axioms have been formulated, they are nevertheless ar- important question of descriptive set theory: PD yields an- ranged in a linear hierarchy of increasing strength. As the swers to all the classical questions about projective sets and name suggests, these axioms assert the existence of cardinal ADL(R) yields answers to all the questions about the impor- numbers with properties implying great size. Such axioms tant larger structure L(R). (See chapter 6 of ref. 2.) seem a fairly natural way ofextending the ZFC axioms. (One The success of determinacy axioms led to a revised pro- might regard the standard axioms of infinity and replace- gram for doing descriptive set theory based on large cardinal ment as the beginning of the hierarchy of large cardinal axi- axioms: Show that large cardinal axioms imply determinacy oms.) axioms. Martin (3) achieved the first step in this revised pro- Do the large cardinal axioms help with the otherwise unan- gram by proving that the determinacy of IV sets follows swerable questions of set theory? On the negative side, they from the existence of a measurable cardinal. Nine years lat- do not settle the continuum hypothesis. Nevertheless, they er, Martin (4) proved the determinacy of II2 sets from a very do imply certain special cases of the generalized continuum strong large cardinal axiom (much stronger than the exis- hypothesis, and they do provide answers to certain ques- tence of supercompact cardinals). In 1984, W. H. Woodin tions ofdescriptive set theory. (See sections 33 and 37 ofref. proved PD and indeed ADL(R) from an even stronger large 1.) For example, Solovay proved (see page 548 of ref. 1) that cardinal axiom, and the program seemed complete. all ZI sets are Lebesgue measurable, assuming the existence This was, however, not quite the case. Though the theo- of a measurable cardinal. (A set is Z1 if it is the continuous rem on IV determinacy was fully satisfactory, there were image of a Borel set; H1 if it is the complement of a I' set; serious questions about the hypotheses of the proofs of II2 Zn+1 if it is the continuous image of a Hn set; An if it is both determinacy and ADL(R). These hypotheses were not well- Zn and Hn1; projective if it is In for some n. For definitions understood; set theorists had until then found no use for any- pertaining to large cardinals, see ref. 1.) Other descriptive thing so strong. To some they seemed not sufficiently re- set-theoretic questions were answered using measurable car- moved from a large cardinal hypothesis proved inconsistent dinals. The hope arose in the late 1960s that large cardinal by K. Kunen. And even without doubting their consistency, axioms might suffice for solving all important problems of one might doubt their necessity. Perhaps substantially weak- descriptive set theory. It was known that measurable cardi- er axioms would suffice to prove HI determinacy and nals were insufficient for this, but Solovay introduced the ADL(R) larger supercompact cardinals and suggested that they might Godel's second incompleteness theorem implies that one suffice. cannot prove the consistency of a large cardinal axiom; one There was little additional immediate success in this pro- can, however, give evidence. Perhaps the strongest evidence gram ofusing large cardinal axioms to investigate descriptive comes from the inner model program. The goal of this pro- gram is to associate to each large cardinal axiom A a canoni- The publication costs ofthis article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" Abbreviations: ZFC, Zermelo-Fraenkel, with the axiom of choice; in accordance with 18 U.S.C. §1734 solely to indicate this fact. PD, projective determinacy; AD, axiom of determinacy. 6582 Downloaded by guest on September 27, 2021 Mathematics: Martin and Steel Proc. NatL Acad Sci USA 85 (1988) 6583 cal inner model MA that is in some sense minimal and whose faith that such a theorem could be proved and with exactly structure can be analyzed in detail. This is not a consistency the correct large cardinal hypothesis. The proof itself is to- proof for A, as one must assume A in order to show that MA tally unrelated to the work of Foreman et al. or of Woodin. satisfies A; still, the full and detailed description of such a (Of course, Woodin's theorem-and so indirectly the work model gives some evidence of consistency. (A hidden incon- of the other authors-is needed to get ADL(R) from our theo- sistency in A should emerge quickly in the theory of MA.) rem.) The technical ideas in our proof have their origin rath- Inner model theory is also useful for showing the necessity er in Mitchell's and later Steel's work on the inner model of large cardinal hypotheses. Usually one of the first theo- program. rems about MA is that in it the reals admit a well-ordering Recently we have carried out the inner model program far definable in a certain form, so that the determinacy of games enough to show that our results on PD are essentially opti- definable in a corresponding form TA fails in MA. Thus A mal. More recently, Woodin has by a different method does not imply rA determinacy. (With more work, one can proved for HIl determinacy and for ADL(R) what are essen- sometimes go on to show TA determinacy equivalent to the tially equivalences of the kind mentioned earlier. existence of a "sharp" of MA-i.e., of a real coding a blue- print for building MA. Such equivalences, which may exist Homogeneous Trees throughout the large cardinal and determinacy hierarchies, expose more fully the connection between the two.) A tree T on a set X is a subset of `0X (i.e., a set of finite The existence of a A3 well-ordering of the reals implies the sequences of elements of X) such that whenever s extends t failure of H1 determinacy; the existence of a Ai well-order- and s E T then t E T. When we study trees Ton products X ing does not. Thus to show the hypothesis of Martin's H2 x Y, we shall think of elements of T as pairs of finite se- proof optimal, one wanted, for each large cardinal axiom A quences (of the same length) rather than as finite sequences weaker than the hypothesis of the proof, an inner model MA of pairs. with a A3 well-ordering of its reals. One also wanted an inner If T is a tree on X, [T] is the set of all infinite branches of model for the hypothesis itself, as an assurance of consisten- T, i.e., {fIf: w- Xand (Vn)f t n E T}. IfA is a subset ofB cy. With the appearance in 1978 of Mitchell (5) (extending x C, then pA = {bI(3c)(b, c) E A}. If K is a cardinal number, the work of Silver, Kunen, and Mitchell), the inner model a subset A of WZ is K-Souslin ifA = p[T] for some tree Ton Z program seemed on the verge of completing these tasks. But, X K. despite great effort, in the next 6 years the theory moved Let T be a tree on Z x X. T is homogeneous if there is a only a little beyond the models of ref. 5. The models con- system (,'Is E subsets of T5, and with In 1984, Foreman et al. (6) proved, assuming the consis- .s(Ts)= 1; tency of supercompact cardinals, the consistency of a num- (ii) the A, are compatible: if s, C s2 (s2 extends sj) then ber of set-theoretic assertions which-according to received 'U1(Y) = 1 * lA2({tlt llh(sl) E Y}) = 1, where lh(sl) is the opinion-should have needed the consistency of much larg- length of sj; er cardinals than supercompact. Woodin (personal commu- (iii) ifx E p[T] then the ultrapower by (p, trhIn E w) is well- nication) then proved-using the techniques of ref. 6, the founded. This ultrapower Mx is defined as follows: Letj, : V following outright theorem: If a supercompact cardinal ex- M, be the elementary embedding associated with the ul- ists, then every projective set of reals is Lebesgue measur- trapower by A,' (We ambiguously use "M3" for the ultra- able. Shelah and Woodin strengthened the conclusion to power and for the transitive class isomorphic to it; we also cover all sets in L(R). Thus supercompact cardinals are in- do not pay attention notationally to the difference between compatible not only with A3 well-orderings of the reals but models and their domains, especially when the interpretation even with there being a well-ordering of the reals that be- of "8" is membership.) For s, 5 S2, define the elementary longs to L(R). embeddingjj1,2 M51 -3 MS2 by jS1S2 ([f]/,1) = f'],5,2, where Shelah and Woodin gradually weakened the hypotheses of f'(t) = f(t t lh(sl)). Forx E `Z let (Mxs, (Jxt,xIn E w)) be the their results. In particular, Woodin brought the hypothesis of direct limit of the system ((MxtIn C £), (ixtnfmIn < m E the measurability of all £1 sets down approximately to the w)). (Of course, Mx is not literally an ultrapower.) When Mx existence of what is now called a Woodin cardinal. A cardi- is well-founded, we shall identify it with the transitive class nal 8 is Woodin if, for every f: 6 -* 8 there are (i) a K < 8 isomorphic to it. We shall use the term Ult(V;(P, Inn 8cw)), closed under f and (ii) an elementary embeddingj: V - M, which is often reserved for the transitive class isomorphic to with M a transitive class, K the least ordinal moved byj, and a well-founded ultrapower, even for non-well-founded ultra- V(j(p)(K) E: M. It is fairly easy to see that all supercompact powers. cardinals are Woodin. In fact, Woodinness is much weaker Note that the converse of iii always holds: Ifx 0 p[T] then than supercompactness. T(x) = U, Txrn has no infinite branch, while (jUrn,X([identi- The work of Foreman et al., Woodin, and Shelah and ty],'r.)In 8 £0) is an infinite branch ofjx(T(x)), where jx = Woodin thus did two things for the determinacy question: (i) J0,x It made it very likely that large cardinal axioms weaker than A tree is K-homogeneous if some system of K-complete the assertion that supercompact cardinals exist mipht imply (i.e., (transitive set such that Y c Vj(K) n M. For each being homogeneously Souslin up the projective hierarchy. finite a C Y, we define a measure Ea by That is, suppose that A C Asw X Zw and that A is homoge- neously Souslin. (Literally, we should construe A as a subset Ea(X) = 1 4*j1 trj(a) E (X). of '(w x w).) Let B = {xJ(Vy C 'cw)(x, y) 0 A}. We would like to prove-from some large cardinal hypothesis-that B is Ea thus concentrates on functions with domain a and range homogeneously Souslin. The following construction, which C VK. Ea is K-complete. Since a is finite Ea is essentially a generalizes that of ref. 8, achieves half of this. (See ref. 7.) measure on VK If a C b, then Let T be a tree on (co X ai) x Z for some set Z, and let (Ls,r)ls, r C 'ho & lh(s) = Mh(r)) witness that T is homoge- Ea(X) = 1 X Eb({f t a C X}) = 1. neously Souslin. Let ro, rl, . . . enumerate <"Iw so that each if sequence is enumerated before its proper extensions. we can We now define a tree T on w X ON, where ON is the class Thus, as with our systems of homogeneity measures, of all ordinal numbers. Let form the ultrapower Ult(V; E) by E = (Ea a E < [YI). There is an elementary embedding k of this ultrapower into M: Set k([F]) = (j(F))(j' tj(a)) for F: a(VK) -3 V. It is easy (s, (ace, * * . alh(s)-l)) C X to check that k Y is the identity and thatj = k o iE, where iE: (i(ViVi2)[(i1 < i2 < lh(s) & C ri2) Ve- Ult(V; E) is the canonical elementary embedding. Thus ri, Ult(V; E) is well-founded. => ai2 al)] By an extender E with crit(E) = K and support(E) = Y, let us mean the E derived from somej as above. (Extenders can LEMMA 1.3. For any x E '0w and any vy - (21zl)+ be defined directly, without reference to an embeddingj.) Recall that 8 is Woodin if for all f: 8 -+ 8 there is a K < 8 x E p[T] x E p[T rv]y (Vy E 1w)(x, y) 0 p[T], with K closed under f and there is an elementaryj: V -> M with crit(j) = K and V(j(f)(K) E M. The last clause can be where T vy = {(s, t)j(s, t) E T & t E 'd^y}. replaced by the following: there is an extender E E Via with Thus, ifA and B are defined from T as above, then B is y crit(E) = K and support (E) D V(Ew))(K). Souslin for each y 2 (21Z1)+. Suppose M and N are transitive proper class models of THEOREM 1. Let T be 8+-homogeneous with 8 a Woodin ZFC. Let K be an ordinal number and suppose that VK+l n M cardinal. Let K < 8. T is K-homogeneous; i.e., all sufficiently = v,+i n N. Let E E M with M satisfying "E is an extender large T r p are K-homogeneous. and crit(E) = K." Since the measures of E are essentially COROLLARY. If there are n Woodin cardinals and a mea- measures on VK, we can form the ultrapower Ult(N; E) of N surable cardinal larger than all of them, then Determina- by E. If Ult(N; E) is well-founded (automatically the case if cy(lln+1)- M is countably closed), then we identify it with the isomor- Proof: Let An < An_1 < ... < A1 be Woodin. Let AO > A1 phic transitive class and denote the corresponding elemen- be measurable. Let An+1 = w. By Lemma 1.2, every 1ll set is tary embedding by iEN. Note that, since VK+l n M = VK+, n AO-homogeneously Souslin and so Al-homogeneously Sous- N, then iEm(K) = iE(K) and ViM'(K) n Ult(M; E) = Vi N,) n lin. By induction, using Theorem 1, it follows that every II Ult(N; E). set is Ak-homogeneously Souslin, for all k c n + 1. For 0 < a ' c, an iteration tree on V of length a is a THEOREM 2. (W. H. Woodin, California Institute of Tech- system nology, personal communication; see ref. 10 for proof from supercompact cardinals). Suppose there are infinitely many (<, (Mk k < a), (Ek k + 1 < a), (Pk k + 1 < a)) Woodin cardinals 8o < 8X < ... with a measurable cardinal greater than all of them. Then every set in L(R) is pB for with the following properties: some B that is &,-homogeneously Souslin for all i. (i) (a, <) is a tree ordering with 0 <-least and with n < m COROLLARY. The hypotheses of Woodin's theorem imply > n < m. ADL(R) (ii) Each Mk is a transitive proper class model of ZFC and The is1.s2 associated with a homogeneous tree suggest the MO= V. following definition: A subset A of 0cw can be represented in (iii) (Pk k + 1 < a) is a nondecreasing sequence of ordi- embedding normal form if there are (M, : s E <'w) and nals. (JS,S21s1 5S2 & Sb, S2 E < 'w) such that each Ms is a transitive (iv) k1 c k2 < a -> n mkl = n Mk2. class, M0 = V, each j3SS2 :MS1 M. is an elementary em- Vpk,+1 Vpkj+1 bedding, the JS1,s2 commute, and, for ah x E "ow, x E A 0. Suppose also that M. satisfies jects in the models on different branches of the trees. "E is an extender and support(E) D Vp+1." Suppose finally LEMMA 3.2 (One-step lemma). Let M and N be transitive that fi s n and crit(E) ' pN. There is a unique extension proper class models ofZFC. Assume that 8 is Woodin in M and inaccessible in V. Let K < 8, let q < 8, let /8 and A' be -'==c,(MkIk en Then Mb is well-founded. In our applications, M will be the Mk of an iteration tree Note that, ifb' #& b is an infinite branch of 9, then the en, n we are building, K will be crit(Ek), E will be Ek, N will be E b' - b witness that Mb' is ill-founded. Lemma 2.2 will be Mk*, and Ult(N; E) will be Mk+1. The well-foundedness of proved in our paper on inner models. The slightly easier Ult(N; E) will follow from Lemma 2.1. countably closed case will be done in our detailed paper on PD. The Main Construction Reflecting Cardinals From now on assume that 8 is Woodin, that T is a tree on (a X co) x Afor some A > 8, and that (1sr) s, r 8 <"co & lh(r) In order to use Woodin cardinals to build iteration trees, we - lh(s)) witness that T is 8+-homogeneous. formulate in this section an equivalent of Woodinness and We wish to use Lemma 3.2 to build iteration trees. It is not use it to obtain the technical lemma that is our principal tool hard to see how to use that lemma and Lemmas 2.1 and 3.1 in building iteration trees. to build iteration trees of finite lengths. But the fact that, in For the remainder of the paper assume that 8 is an un- Lemma 3.2, we must have e < / gives us problems in build- countable strong limit cardinal. ing iteration trees of infinite length. To circumvent this prob- For ordinals a and / with a < 8 and for z E `'(Vs+p), the lem, let A < c0 < cl < c2 be cardinals such that (i) all Ck, k < (a, (3)-type of z is the set of all formulas q(v) of the language 2, are strong limit cardinals of cofinality greater than 8 and of set theory with additional constants for 8 (unless /3 = 0) (ii) co and cl satisfy the same formulas in Vc2, allowing pa- and elements of Va such that Vt+s (qzz]. rameters from VA+1. Remark: For infinite a, the (a, /B)-type ofz is essentially a Condition ii will guarantee that, for our purposes, the cy- subset of Vt,. Thus if co - A < a and p < /3 then the fact that T cle c0 + 1, co - cl, c0 + 1, etc., will act as an infinite de- is the (A, p)-type of z is expressed by a member of the (a, /3)- scending sequence of ordinals. type of z_(p). Let ro, rl, r2, . . . be as in the definition of T. Let < be If K < 8 and z E <'(V8+8), then K is (-reflecting in z if for defined by 2m < 2n * m < n; 2m + 1 < 2n + 1 t rm+l1 all a < 8 there is an extender E in V, with crit(E) = K and r,+1; otherwise m < n X 0 = m < n. support(E) 2 Va such that the (a, /3)-type of z in V is the With each s < 'co, we shall associate an iteration tree 3, same as the (a, iE(,/))-type of iE(Z) in Ult(V; E). - (c< 21h(s) + 1, (Mk(s) k s 21h(s)), (Ek(s) k < 21h(s)), Remark: If A < a and p < /3, the fact that A is p-reflecting (pk(s) k < 21h(s))) with Os E V8. For s 5 s', 0s will be in z is expressed by a member of the (a, /3)-type of zp). extended, in the obvious sense, by 0-sT. LEMMA 3.1. Assume that 8 is Woodin. For all /3 and all z For x E 'wco, we shall thus have an iteration tree 9 (x) = E <'(Vt,+,O), the set of K that are /3-reflecting in z is un- (Mk(x) k 8 a), (Ek(x) k 8 a), (pk(x) k 8 a)), obtained by bounded in 8. putting together the 9, f,. The branches of 9, will be Even To prove Lemma 3.1, assume that a(K) witnesses that K is = the set of all even natural numbers and, for each y 8 Zeo, by not /-reflecting in z for each K with A < K < 8. Apply Wood- = {0} U {2k + 1 | rk+1 C y}A inness to f where f(K) = A + a(K) + co, getting E with crit(E) With each s 8 <'co we shall also associate a finite se- = K*. Show that the initial segment of E with support quence u(s). If k = lh(s), then lh(u(s)) will be lh(rk) and u(s) V(iE(a)XK*) gives a contradiction in Ult(V; E). will belong to (io,2-l(s))(T(s rlh(rt),rk)) whenever lh(s) > 0. Lemma 3.1 will allow us to build iteration trees and, as we (Here imn(5), for m < n c 21h(s), is the canonical elementary build, to preserve and extend correspondences between ob- embedding associated with ?Ts.) Furthermore we shall have Downloaded by guest on September 27, 2021 6586 Mathematics: Martin and Steel Proc. NatL Acad Scl. USA 85 (1988) that (U2k-1,2k'1(S '))(U(S)) C u(s') whenever s C s', 0 < k = (s))(t) in M2jh(s)(s) is the same as the (K(s), co + 1)-type of lh(s), k' = lh(s'), and rk C rk'. This means that the images of ((io,21h(s))(s))(T)) u(s) in M2,h(S))_1(s) when lh(s) > 0; the u(x t k) for 2k - 1 E by give an infinite branch of the (ii) K(S) is (e(s, t) + l)-reflecting in ((io,21h(s)(S))(T)) image of T((x, y)) in Mby. If(x, y) 0 p[T], this implies that Mb (io,21h(s))(t) in M2,h(s)(5); is ill-founded. If (Vy)(x, y) 0 p[T] then the ranks of the (rk, (iii) u(s) is definable in Vc0+l nM2h(s)-1(s) from elements u(x [ k)) in the well-founded trees io,2k-1 (T(x)) satisfy the of Vs U {8, (io,2h(s)_1(s))(T)} when lh(s) > 0. hypotheses ofLemma 2.2, so that MEVn(X) is well-founded. The induction step of the construction proceeds by two In summary, we have the following: If (Vy) (x, y) 0 p[T], applications of Lemma 3.2 for each t E Xs, using co and cl as then MEven(X) is well-founded. indicated above, and using.the 8+-completeness of the 1SA,> For an embedding normal form, we also want to make sure and statement iii above to obtain u(s) and 9s independent of t that MEVfn(x) is not well-founded if (3y)(x, y) E p[T]. To this for t belonging to a measure one set Xs. We omit here the end we shall define, for each s E Logic Colloquium '78, eds. Boffo, M., e, Van Dalen, D. & McAloon, K. (North-Holland, Amsterdam), represented by e(s, t) in the ultrapower by gas Ilh(rk) r), where pp. 303-316. k = lh(s). s = (i2nlh(s)(es |) n < lh(s)) can be shown to 6. Foreman, M., Magidor, M. & Shelah, S. (1988) Ann. Math. belong to (io,2h(s)(s)(T,). Let vs(X) = 1 <* Ts E (io,2,h(s)(s))(X). 127, 1-48. (To make these measures K-complete, just choose crit(E0(s)) 7. Kechris, A. S. (1981) in Cabal Seminar 1977-79, Lecture > K.) Notes in Mathematics, eds. Kechris, A. S., Martin, D. A. & The construction of the ?s, u(s), Xs, and e(s, t) proceeds Moschovakis, Y. N. (Springer, Berlin), pp. 33-74. by induction on Wh(s). We also define ordinals K(s) that will 8. Martin, D. A. & Solovay, R. M. (1969) Ann. Math. 89, 138- be the common critical points of the E2,k(s') for all k and s' 159. such that s' Q s, Wh(s') > k, lh(rk) = lh(rih(s)) + 1, and rk D 9. Dodd, A. (1982) The Core Model, London Mathematical Socie- In ty Lecture Notes 61 (Cambridge Univ. Press, Cambridge, En- rlh(s). addition to the properties already stated, we shall gland). have 10. Woodin, W. H. (1988) Proc. Natl. Acad Sci. USA 85, 6587- (i) the (c(s), e(s, t) + 1)-type of ((io,2lh(s)(S))(T)) (io,21h(s) 6591. Downloaded by guest on September 27, 2021