Submitted on October 23, 2015 to the Notre Dame Journal of Formal Logic Volume ??, Number ??,

A Long Pseudo-comparison of Premice in L[x] Farmer Schlutzenberg

A significant open problem in inner model theory is the analysis of HODL[x] as a strategy premouse, for a Turing cone of reals x. We describe here an obstacle to such an analysis. Assuming sufficient large cardinals, for a Turing cone of reals x there are premice M, N in L[x], and countable in L[x], such that the pseudo-comparison of L[M] with L[N] succeeds, is in L[x], and lasts exactly L[x] ω1 stages. Moreover, we can take M = M1 |(δ + )M1 where M1 is the minimal iterable proper class inner model with a Woodin cardinal, and δ is that Woodin. We can take N such that L[N] is M1 -like and short-tree-iterable. Abstract

1 Introduction

A central program in descriptive inner model theory is the analysis of HODW , for transitive models W satisfying ZF + AD+ ; see [8], [6], [2], [5]. For the models W for which it has been successful, the analysis yields a wealth of information regarding HODW (including that it is fine structural and satisfies GCH), and in turn about W . Assume that there are ω many Woodin cardinals with a measurable above. A primary example of the previous paragraph is the analysis of HODL(R) . Work of Steel and Woodin showed that HODL(R) is an iterate of Mω augmented with a fragment of its iteration strategy (where Mn is the minimal iterable proper class inner model with n Woodin cardinals). The addition of the iteration strategy does not add reals, and so the ODL(R) reals are just R ∩ Mω . The latter has an analogue for L[x], which has been known for some time: for a cone of reals x, the ODL[x] reals are just R ∩ M1 . Given this, and further analogies between L(R) and L[x] and their respective HODs, it is natural to ask whether there the full HODL[x] is an iterate of M1 , adjoined with a fragment of its iteration strategy. Woodin has conjectured that this is so for a cone of reals x; for a precise statement see [3, 8.23]. Woodin has proved approximations to this conjecture. He analyzed HODL[x,G] , for a cone of reals x, and G ⊆ Coll(ω, < κ) a generic filter over L[x], where κ is the least inaccessible of L[x]; see [3, 8.21] and [2]. However, the conjecture regarding HODL[x] is still open. 2010 Mathematics Subject Classification: 03E45 Keywords: Inner model, Ordinal definable, Comparison

1

2

F. Schlutzenberg

In this note, we describe a significant obstacle to the analysis of HODL[x] . 1.1 Background We give a brief summary of some relevant definitions and facts. We

assume familiarity with the fundamentals of inner model theory; see [8], [4]. One does not really need to know the analysis of HODL[x,G] , but familiarity does help in terms of motivation; the system F described below relates to that analysis. We do rely on some smaller facts from [2, §3]. Let us give some terminology, and recall some facts from [2]. We say that a premouse N is pre-M1 -like iff N is proper class, 1-small, and has a (unique) Woodin cardinal, denoted δ N . (The notion M1 -like of [2] makes further demands.) Let P, Q be pre-M1 -like. Given a normal iteration tree T on P, T is maximal iff lh(T ) is a limit and L[M(T )] has no Q-structure for M(T ) (so L[M(T )] is pre-M1 -like with Woodin δ (T )). A premouse R is a (non-dropping) pseudo-normal iterate of P iff there is a normal tree T on P such that either T has successor length and R = M∞T , the last model of T (and [0, ∞]T does not drop), or T is maximal and R = L[M(T )]. A premouse R is a pseudo-iterate of P iff there is n < ω and (R0 , R1 , . . . , Rn ) such that R0 = P and Rn = R and each Ri+1 is pre-M1 like and is a pseudo-normal iterate of Ri . A pseudo-comparison of (P, Q) is a pair (T , U ) of normal padded iteration trees of equal length, formed according to the usual rules of comparison, such that either (T , U ) is a successful comparison, or either T or U is maximal. A (z-)pseudo-genericity iteration is defined similarly, formed according to the rules for genericity iterations making a real (z) generic for Woodin’s extender algebra. We say that P is normally short-tree-iterable iff for every normal, non-maximal iteration tree T on P of limit length, there is a T -cofinal wellfounded branch through T , and every putative normal tree T on P of length α + 2 has wellfounded last model (that is, we never first encounter an illfounded model at a successor stage). If P|δ P ∈ HCL[x] , then normal short-tree-iterability is absolute between L[x] and V . If P, Q are normally short-tree-iterable then there is a pseudo-comparison (T , U ) of (P, Q), and if T has a last model then [0, ∞]T does not drop, and likewise for U . By Turing determinacy we mean the statement that every set of Turing degrees either contains or is disjoint from a cone. 1.2 The HOD of L[x] It has been suggested1 that one might analyze HODL[x] using

an ODL[x] directed system F such that: – the nodes of F are pairs (N, s) such that s ∈ OR<ω and N is a normally short-tree iterable, pre-M1 -like premouse with N|δ N ∈ HCL[x] , – for (P,t), (Q, u) ∈ F , we have (P,t) ≤F (Q, u) iff t ⊆ u and Q is a pseudoiterate of P, and – (M1 , 0) / ∈ F. If such systems existed, satisfying some further requirements regarding the sets s, strengthening the iterability requirements, and including countable directedness (for each fixed s), then there would have been a reasonable scenario for analyzing HODL[x] , making use of Neeman’s genericity iterations.2 The primary difficulty in analyzing HODL[x] in this manner is in arranging that F be directed, even finitely. For this, it seems most obvious to try to arrange that F be closed under pseudo-comparison of pairs. However, we show here that, given sufficient large cardinals, there is a cone of reals x such that if F is as above, then F is not closed under pseudo-comparison of pairs.

A Long Pseudo-comparison

3

The proof proceeds by finding a node (N, 0) / ∈ F such that, letting (T , U ) be the pseudo-comparison of (M1 , N), then T , U are in fact pseudo-genericity iterations L[y] L[z] L[x] of M1 , N respectively, making reals y, z ∈ L[x] generic, where ω1 = ω1 = ω1 . Letting W be the output of the pseudo-comparison, we will have W |δ W ∈ L[x], so W [z] L[x] L[x] ω1 = ω1 , which implies that δ W = ω1 , so (W, 0) / ∈ / F . We now proceed to the details. 2 The Pseudo-comparison

For a formula ϕ in the language of set theory (LST), ζ ∈ OR, and x ∈ R, let Axϕ,ζ be the set of all M ∈ HCL[x] such that L[x] |= ϕ(ζ , M), and L[M] is a normally shorttree-iterable pre-M1 -like premouse and M = L[M]|δ L[M] . Note that ϕ does not use x as a parameter. So by absoluteness of normal shorttree-iterability (between L[x] and V , for elements of HCL[x] ), Axϕ,ζ is ODL[x] . So Axϕ,ζ is a collection of premice like those involved in the system F (restricted to their Woodins). Assume Turing determinacy and that M1# exists and is fully iterable. Then for a cone of reals x, for every formula ϕ in the LST and every ζ ∈ OR, if M1 |δ M1 ∈ Axϕ,ζ then there is R ∈ Axϕ,ζ such that the pseudo-comparison of M1 with Theorem

L[x]

L[R] has length ω1 . Suppose not. Then we may fix ϕ such that for a cone of x, the theorem fails for ϕ, x. Fix z in this cone with z ≥T M1# . Let W be the z-genericity iteration on M1 (making z generic for the extender algebra), and Q = M∞W . By standard arguments (see [2]), Q[z] = L[z], L[z] lh(W ) = ω1 + 1 = δ Q + 1, Q|δ Q = M(W  δ Q ), and T =def W  δ Q is the z-pseudo-genericity iteration of M1 , and T ∈ L[z]. Let B be the extender algebra of Q and let P be the finite support ω-fold product of B. For p ∈ P and i < ω let pi be the ith component of p. Let G ⊆ P be Q-generic, with z0 = z where x =def hzi ii<ω is the generic sequence of reals. Then Proof

Q[G] = Q[x] = L[x] and x >T z. Let ζ ∈ OR witness the failure of the theorem with respect to ϕ, x. So M1 |δ M1 ∈ Axϕ,ζ . L[x]

By [1, Lemma 3.4] (essentially due to Hjorth), P is δ Q -cc in Q, so δ Q ≥ ω1 , L[z] L[x] but δ Q = ω1 , so δ Q = ω1 . So it suffices to see that there is some R ∈ Axϕ,ζ such that the pseudo-comparison of M1 with L[R] has length δ Q . For e ∈ ω and y ∈ R let Φye : ω → ω be the partial function coded by the eth Turing program using the oracle y. Let e ∈ ω be such that Φze is total and codes M1 |δ M1 . Let x˙ be the P-name for x, and for n < ω let z˙n be the P-name for zn . Let p ∈ G be such that p

Q P

ψ(˙z0 ), where ψ(v) asserts “Φve is total and codes a premouse R ∈ Ax˙ˇ

ϕ,ζˇ

such

that the v-pseudo-genericity iteration of L[R] produces a maximal tree U of length ˇ δˇQ ”. In the notation of this formula, δˇQ with M(U ) = L[E]| p

Q P

“R ∈ / Vˇ ”, because p

Q P

“E0U ∈ / M(U )”.

F. Schlutzenberg

4

By genericity, we may fix q ∈ G such that q ≤ p and for some m > 0, qm = q0 . Note Q

that q P ψ(˙zm ). Let R˙ i be the P-name for the premouse coded by Φze˙i (or for 0/ if this does not code a premouse). Also let z˙00 , z˙01 be the B × B-names for the two B × B-generic reals (in z˙0 order), and let R˙ 0i be the B × B-name for the premouse coded by Φei . We may fix r ≤ q, r ∈ G, such that r

Q P

“R˙ 0 6= R˙ m ”.

For otherwise there is r ≤ q, r ∈ G, such that r

Q ˙ P “R0

(1) = R˙ m ”. But since

/ Q, M1 |δ M1 = R˙ G 0 ∈ there are s,t ∈ B, s,t ≤ r0 , such that Q B×B

(s,t)

“R˙ 00 6= R˙ 01 ”.

Therefore there are u, v ∈ B, with u ≤ r0 and v ≤ rm , such that Q B×B

(u, v)

“R˙ 00 6= R˙ 01 ”.

Let w ≤ r be the condition with wi = ri for i 6= 0, m, and w0 = u and wm = v. Then Q P

w

“R˙ 0 6= R˙ m ”,

a contradiction. M1 and R ∈ Ax Q So letting R = R˙ G m , we have R 6= M1 |δ ϕ,ζ and Q|δ = M(U ), where Q U is the zG m -pseudo-genericity iteration of L[R], and lh(U ) = δ . We defined T ∗ ∗ earlier. Let T , U be the padded trees equivalent to T , U , such that for each ∗ ∗ ∗ ∗ ∗ ∗ α, either EαT 6= 0/ or EαU 6= 0, / and if EαT 6= 0/ 6= EαU then lh(EαT ) = lh(EαU ). Let (T 0 , U 0 ) be the pseudo-comparison of (M1 , L[R]) (recall that L[R] is normally short-tree iterable as R ∈ Axϕ,ζ ). We claim that (T 0 , U 0 ) = (T ∗ , U ∗ ); this completes the proof. For this, we prove by induction on α that (T 0 , U 0 )  (α + 1) = (T ∗ , U ∗ )  (α + 1). This is immediate if α is a limit, so suppose it holds for α = β ; we prove it for ∗ ∗ α = β + 1. Let λ = lh(EβT ) or λ = lh(EβU ), whichever is defined. Because ∗

∗

M(T ∗ ) = Q|δ Q = M(U ∗ ), the least disagreement between MβT and MβU has index ∗

∗

≥ λ , so we just need to see that EβT 6= EβU . ∗

∗

So suppose that EβT = EβU . In particular, both are non-empty. Then there is Q

s ∈ G such that s ≤ r (see line (1)) and s P “For i = 0, m, let Ti be the z˙i -pseudogenericity iteration of L[R˙ i ]. Then T0 and Tm use identical non-empty extenders E of index λˇ .” Because Q s P ψ(˙z0 ) & ψ(˙zm ), Q ∗ ˇ λˇ , but E ∈ also s P “Letting E be as above, E ⊆ L[E]| / Vˇ ”; here EβT ∈ / Q because λ G G is a cardinal of Q. But since Ti is computed in Q[zi ] (for i = 0, m) we can argue as before (as in the proof of the existence of r as in line (1)) to reach a contradiction. 

A Long Pseudo-comparison

5

A slightly simpler argument, using B × B instead of P, proves the weakening of the theorem given by dropping the parameter ζ . The author does not see how to prove the full theorem using B × B instead of P. This is because in the argument given, ζ depends on x, and the choice of the conditions p, q depend on ζ .3 Notes

1. For example, at the AIM Workshop on Descriptive inner model theory, June, 2014. L[x]

2. Woodin’s genericity iterations produce trees of length ω1 . Moreover, it seems that HCL[x] need not be sufficiently closed under the existence of collapse generics to allow an obvious analysis of HODL[x] using Neeman’s genericity iterations. (Thanks to John Steel for pointing out an error that appeared in a draft of this paper, regarding this point.) But it seems this issue could have been avoided by using the following fact, due to Steel, together with related calculations: Assume (enough) determinacy. Then for a cone of x, L[x] and L[x, y] have the same theories in ordinal parameters whenever y is Cohen over L[x]. (The proof, which we include with Steel’s permission, is as follows. For reals x, let [x] denote the Turing degree of x. Let F([x]) be the least (ϕ, α) such that α ∈ OR and L[x] |= ϕ(α) but for some Cohen-generic y over L[x], L[x, y] |= ¬ϕ(α), if such (ϕ, α) exists; let F([x]) = 0 otherwise. Then F([x]) = 0 for a cone of x, because otherwise the proof of [7, 4.3] leads to contradiction. There is also an alternate proof using M1# : We may assume that L[x] = N[G], where N is some non-dropping iterate of M1 , and G is (N, P)generic where P = Coll(ω, δ N ), and so L[x, y] = N[G][H], for some (N[G], P)-generic H. It follows that L[x] and L[x, y] have the same theories in ordinal parameters. Clearly both proofs work with much less than full determinacy, but the former proof does not require M1# .) 3. So if one tries to run the same argument but with B × B instead of P, one must first choose a generic pair of reals x = (z0 , z1 ), thus determining ζ , but then even if we had tried to be selective about z1 , it seems there might not be any q ∈ G analogous to that found in the proof using P. On the other hand, if there is no parameter ζ involved, we can be selective enough about z1 .

References

[1] Farah, I., “The extender algebra and Σ21 -absoluteness,” To appear in Cabal Seminar: Volume 4. Available on author’s website. 3 [2] John R. Steel, and W. Hugh Woodin, “HOD as a core model; Ordinal Definability and Recursion Theory: The Cabal Seminar, Volume III,” pp. 257–346 Cambridge University Press, 2016, URL http://dx.doi.org/10.1017/CBO9781139519694.010. Cambridge Books Online. 1, 2, 3 [3] Koellner, P., and W. H. Woodin, “Large cardinals from determinacy,” pp. 1951– 2119 in Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, URL http://dx.doi.org/10.1007/978-1-4020-5764-9_24. Zbl 1198.03072. MR 2768702. 1 [4] Mitchell, W. J., and J. R. Steel, ume 3 of Lecture Notes in Logic,

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http://dx.doi.org/10.1007/978-3-662-21903-4. MR 1300637 (95m:03099). 2

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[5] Sargsyan, G., Hod Mice and the Mouse Set Conjecture, volume 1111 of Memoirs of the American Mathematical Society Series, American Mathematical Society (AMS), Providence, RI, 2015. 1 [6] Steel, J. R., “HODL(R) is a core model below Θ,” Bull. Symbolic Logic, vol. 1 (1995), pp. 75–84. URL http://dx.doi.org/10.2307/420947. Zbl 0826.03022. MR 1324625 (97a:03059). 1 [7] Steel, J. R., “Derived models associated to mice,” pp. 105–193 in Computational prospects of infinity. Part I. Tutorials, volume 14 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., Hackensack, NJ, 2008. Available at author’s website. MR MR2449479. 5 [8] Steel, J. R., “An outline of inner model theory,” pp. 1595–1684 in Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, URL http://dx.doi.org/10.1007/978-1-4020-5764-9_20. Zbl 1198.03070. MR 2768698. 1, 2

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