ORDINAL DEFINABLE SUBSETS OF SINGULAR CARDINALS JAMES CUMMINGS, SY-DAVID FRIEDMAN, MENACHEM MAGIDOR, ASSAF RINOT, AND DIMA SINAPOVA Abstract. A remarkable result by Shelah states that if 휅 is a singular strong limit cardinal of uncountable cofinality then there is a subset x of 휅 such that HODx contains the power set of 휅. We develop a version of diagonal extender- based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in factitis consistent that for some singular strong limit cardinal 휅 of countable cofinality + 휅 is supercompact in HODx for all x ⊆ 휅. 1. Introduction It is a familiar phenomenon in the study of singular cardinal combinatorics that singular cardinals of countable cofinality can behave very differently from those of uncountable cofinality. For example in the area of cardinal arithmetic we may contrast Silver's theorem [11] with the many consistency results producing models where GCH first fails at a singular cardinal of countable cofinality [2]. The results in this paper show that there is a similar sharp dichotomy involving questions about the definability of subsets of a cardinal rather than the size of its powerset. Shelah [12] proved that if 휅 is a singular strong limit cardinal of uncountable cofinality then there is a subset x ⊆ 휅 such that P (휅) ⊆ HODx. In this paper we show that the hypothesis is essential, by proving: Theorem. Suppose that 휅 < 휆 where cf(휅) = !, 휆 is inaccessible and 휅 is a limit of 휆-supercompact cardinals. There is a forcing poset Q such that if G is Q-generic then: ∙ The models V and V [G] have the same bounded subsets of 휅. ∙ Every infinite cardinal 휇 with 휇 ≤ 휅 or 휇 ≥ 휆 is preserved in V [G]. ∙ 휆 = (휅+)V [G]. ∙ For every x ⊆ 휅 with x 2 V [G], (휅+)HODx < 휆. From stronger assumptions we can use Q to obtain a model in which 휅 is a + singular strong limit cardinal of cofinality !, and 휅 is supercompact in HODx for all x ⊆ 휅. Cummings was partially supported by the National Science Foundation, grants DMS-1101156 and DMS-1500790. Friedman would like to thank the Austrian Science Fund for its generous support through the research project P25748. Rinot was partially supported by the Israel Science Foundation, grant 1630/14. Sinapova was partially supported by the National Science Foundation, grants DMS-1362485 and Career-1454945. 1 2 J. CUMMINGS, S.-D. FRIEDMAN, M. MAGIDOR, A. RINOT, AND D. SINAPOVA Acknowledgements. The results in this paper were conceived and proved during three visits to the American Institute of Mathematics, as part of the Institute's SQuaRE collaborative research program. The authors are deeply grateful to Brian Conrey, Estelle Basor and all the staff at AIM for providing an ideal working envi- ronment. Notation. Our notation is mostly standard. We write `(s) for the length of a sequence s. When Q is a forcing poset and q 2 Q we write Q # q for fr 2 Q : r ≤ qg, with the partial ordering inherited from Q. 2. A proof of Shelah's theorem For the reader's benefit we give a short and self-contained proof of the result by Shelah quoted in the Introduction. Let 휅 be a strong limit singular cardinal of uncountable cofinality 휇. Fix a set x ⊆ 휅 such that L[x] contains an enumeration (t휂)휂<휅 of H휅, together with a sequence of ordinals (훼i)푖<휇 which is increasing and cofinal in 휅. To each set X ⊆ 휅 we associate a function fX : 휇 ! 휅, where fX (i) = 휂 for the unique 휂 such that X \ 훼i = t휂. It is easy to see that if X 6= Y then fX (i) 6= fY (i) for all large i. We now impose a strict partial ordering on P (휅) by defining X C Y if and only if fX (i) < fY (i) for all large i; since 휇 = cf(휇) > !, the ordering C is well-founded. Let R훼 be the set of elements of P (휅) with C-rank 훼, and note that if X and Y are distinct elements of R훼 then there are unboundedly many i < 휇 with fX (i) > fY (i), since otherwise fX (i) < fY (i) for all large i and it would follow that X C Y . We 휇 claim that jR훼j ≤ 2 , so that in particular jR훼j < 휅. Supposing for contradiction 휇 that jR훼j > 2 we fix a sequence (Xj)j<(2휇)+ of distinct elements of R훼, and then define a colouring c of pairs from (2휇)+ by setting c(j; j0) = i for the least i < 휇 with fXj (i) > fXj0 (i). By the Erd}os-Radotheorem c has a homogeneous set of order type 휇+, which is impossible because it would yield an infinite decreasing sequence of ordinals. S We now define a well-ordering ≺훼 of R훼. To do this we let A훼 = frange(fX ): X 2 R훼g, note that jA훼j < 휅, and define g훼 to be the order-isomorphism between A훼 and ot(A훼). If X 2 R훼 then g훼 ∘ fX 2 H휅, and we define X ≺훼 Y if and only if the index of g훼 ∘ fX is less than that of g훼 ∘ fY in the enumeration (t휂)휂<휅 of H휅. Forming the sum of the orderings ≺훼 in the natural way, we obtain a well- ordering ≺ of P (휅) which is clearly definable from x. Now every element of P (휅) is ordinal definable from x as \the 휂th element of P (휅) in the ordering ≺", so that P (휅) ⊆ HODx. 3. The main theorem 3.1. Setup. Let 휅 < 휆 with 휆 strongly inaccessible and 휅 = supn 휅n, where (휅n)n<! is a strictly increasing sequence of 휆-supercompact cardinals. Fix Un a 휅n-complete fine normal ultrafilter on P휅n 휆, and for 휅 ≤ 훼 < 휆 let U푛,훼 be the projection of Un to P휅n 훼 via the map x 7! x \ 훼. We need a form of diagonal intersection lemma. Lemma 1. Let k < l < !, let (훼i)k≤i<l be a ≤-increasing sequence of elements of Q [휅, 휆), and let S ⊆ k≤i<l P휅i 훼i be a set of ⊆-increasing sequences. ORDINAL DEFINABLE SUBSETS OF SINGULAR CARDINALS 3 Let m 2 [l; !), let 훼 2 [훼l−1; 휆), and let A~ = (As)s2S be an S-indexed family with As 2 U푚,훼 for each s 2 S. Then the following set is in U푚,훼: D = fx 2 P휅m 훼 : 8(xk; : : : ; xl−1) 2 S [xl−1 ⊆ x =) x 2 A(xk;:::xl−1)]g: Proof. Let j : V ! M be the ultrapower map computed from Um. To show that D 2 U푚,훼 it suffices to show that j\훼 2 j(D). Let t = (yk; : : : yl−1) 2 j(S) with yl−1 ⊆ j\훼. Write B~ = j(A~). We need to show that j\훼 2 Bt. For each i with k ≤ i < l, yi ⊆ j\훼i and jyij < 휅i < 휅m = crit(j), so that yi = j\xi = j(xi) for some xi 2 P휅i 훼i. Write s = (xk; : : : xl−1). Then t = j(s) and hence s 2 S. In particular As 2 U푚,훼, and so j\훼 2 j(As) = Bt. In the setting of Lemma1 we will refer to D as the diagonal intersection of A~ and we will write \s b x" as shorthand for \s = (xk; : : : xl−1) with xl−1 ⊆ x". With a view to the proof of Lemma 12 below we also need a technical lemma about measure one sets: Lemma 2. Let i < ! and let 훼i ≤ 훽i ≤ 훽i+1 be ordinals Let B 2 U푖,훽i and let 0 C 2 Ui+1,훽i+1 . Let C be the set of y 2 C such that for every x 2 B with x \ 훼i ⊆ y 0 0 0 there is x 2 B \ P (y) with x \ 훼i = x \ 훼i. Then C 2 Ui+1,훽i+1 . Proof. Let j : V ! M be the ultrapower map computed from Ui+1. It suffices to 0 prove that j\훽i+1 2 j(C ). Since C 2 Ui+1,훽i+1 , we have j\훽i+1 2 j(C). Suppose that X 2 j(B) satisfies 0 0 X\j(훼i) ⊆ j\훽i+1; we shall find X 2 j(B\j\훽i+1) such that X \j(훼i) = X\j(훼i). Since B 2 P휅i 훽i, 휅i < 휅i+1 = crit(j) and X 2 j(B), it follows that X \ j(훼i) = 0 0 j(t) = j\t for some t 2 P휅i 훼i. By elementarity there is x 2 B such that x \훼i = t. 0 0 0 0 Now j(x ) 2 j(B), j(x ) = j\x ⊆ j\훽i+1, and j(x ) \ j(훼i) = j(t) = X \ j(훼i), so 0 0 X = j(x ) is as required. 3.2. The poset Q. The poset Q is a diagonal extender-based forcing of the general type introduced by Gitik and Magidor [4]. For the experts, we note the main innovations: ∙ The extender used here on level n is formed from the approximations U푛,훼 to the supercompactness measure Un, rather than from measures approxi- mating some short extender. ∙ The supports of the components of Q are quite large. This is necessary because the proofs below require long diagonalisations, which are only pos- sible because the coordinates of the extender on level n are linearly ordered and all the measures U푛,훼 are normal.