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.