The standard comparison lemma of inner model theory is deficient, in that it does not in general produce a comparison of all the relevant inputs. How two mice compare can depend upon which iteration strategies are used to compare them. We shall outline here a method for comparing iteration strategies that removes this defect.

Abstract We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is an absolutely definablecore modelthat is close toVin various ways.

Abstract We show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ ℵ such that □κ and □(κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the generic extension by Col(κ, κ+). Of special interest is 1) with κ = ℵ3 since it follows from PFA by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over Kc∥κ.

Abstract Let L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal κ which is not a limit of Woodin cardinals there is some cutpoint t < κ such that Jκ[E] is iterable above t with respect to iteration trees of length less than κ. As an application we show L[E] to be a model of the following two cardinals versions of the diamond principle. If λ > κ > ω1 are cardinals, then holds true, and if in addition λ is regular, then holds true.

In this note we shall prove Theorem 0.1. Let be a countably ω-iterable -mouse which satisfies AD, and [α, β] a weak gap of . Suppose is captured by mice with iteration strategies in ∣α. Let n be least such that ; then we have that believes that has the Scale Property. This complements the work of [5] on the construction of scales of minimal complexity on sets of reals in K(ℝ). Theorem 0.1 was proved there under the stronger hypothesis that all sets definable over are determined, although without the capturing hypothesis. (See [5, Theorem 4.14].) Unfortunately, this is more determinacy than would be available as an induction hypothesis in a core model induction. The capturing hypothesis, on the other hand, is available in such a situation. Since core model inductions are one of the principal applications of the construction of optimal scales, it is important to prove 0.1 as stated. Our proof will incorporate a number of ideas due to Woodin which figure prominently in the weak gap case of the core model induction. It relies also on the connection between scales and iteration strategies with the Dodd-Jensen property first discovered in [3]. Let be the pointclass at the beginning of the weak gap referred to in 0.1. In section 1, we use Woodin's ideas to construct a Γ-full a mouse having ω Woodin cardinals cofinal in its ordinals, together with an iteration strategy Σ which condenses well in the sense of [4, Def. 1.13]. In section 2, we construct the desired scale from and Σ.

The full-background-extender Kc -construction of [2] has the property that, if it does not break down and produces a final model , then Ή is Woodin in V ⇒ Ή is Woodin in , for all Ή. It is natural to ask whether κ is strong in V ⇒ κ is λ-strong in , for all κ, or even better, κ is λ-strong in V ⇒ κ is λ-strong in . As one might suspect, the more useful answer would be “yes”. For the Kc-construction of [2], this question is open. The problem is that the construction of [2] is not local: because of the full-background-extender demand, it may produce mice projecting to ρ at stages much greater than ρ. Because of this, there is no reason to believe that if E is a λ-strong extender of V, then The natural proof only gives that if κ is Σ2-strong, then Σ, is strong in . We do not know how to get started on this question, and suspect that in fact strong cardinals in V may fail to be strong in , if is the output of the construction of [2]. Therefore, we shall look for a modification of the construction of [2]. One might ask for a construction with output such that (1) iteration trees on can be lifted to iteration trees on V, (2) ∀δ(δ is Woodin ⇒ δ is Woodin in ), and (3) (a) ↾κ(κ is a strong cardinal ⇒ κ is strong in ), and (b) ↾κ↾λ(Lim(λ) Λ κ is λ-strong ⇒ κ is λ-strong in ).

Abstract We produce counterexamples to the unique and cofinal branches hypotheses, assuming (slightly less than) the existence of a cardinal which is strong past a Woodin cardinal.

§1. Introduction. The basic problem of inner model theory is how to construct mice satisfying hypotheses appreciably stronger than “there is a Woodin limit of Woodin cardinals”. We have a family of constructions, the Kc-constructions, which ought to produce such mice under the appropriate hypotheses on V. Perhaps the most important thing we lack is a proof that the countable elementary submodels of premice produced by a Kc-construction are ω1 + 1-iterable. The best partial results in this direction are those of Neeman ([4]) for Kc-constructions making use of full background extenders over V, and those of Andretta, Neeman, and Steel ([1]) for arbitrary Kc-constructions. Let be a countable premouse embedded by π into a level of the Kc-construction ℂ. If ℂ uses only full extenders over V as its background extenders, then π and ℂ enable one to lift an evolving iteration tree on to an iteration tree * on V. (See [3, §12].) The good behavior of * guarantees that of . The natural conjecture here is that V is ω1 + 1-iterable with respect to such trees * by the strategy of choosing the unique wellfounded branch. The open question here is uniqueness, since by [2] the uniqueness of the wellfounded branch chosen by * at limit stages strictly less than λ implies the existence of a wellfounded branch to be chosen at λ.

In this paper we shall prove Theorem 0.1. Suppose there is a singular strong limit cardinal κ such that □κ fails; then AD holds in L(R). See [10] for a discussion of the background to this problem. We suspect that more work will produce a proof of the theorem with its hypothesis that κ is a strong limit weakened to ∀α < κ (αω < κ), and significantly more work will enable one to drop the hypothesis that K is a strong limit entirely. At present, we do not see how to carry out even the less ambitious project. Todorcevic [23] has shown that if the Proper Forcing Axiom (PFA) holds, then □κ fails for all uncountable cardinals κ. Thus we get immediately: It has been known since the early 90's that PFA implies PD, that PFA plus the existence of a strongly inaccessible cardinal implies ADL(ℝ) and that PFA plus a measurable yields an inner model of ADℝ containing all reals and ordinals. As we do here, these arguments made use of Tororcevic's work, so that logical strength is ultimately coming from a failure of covering for some appropriate core models. In late 2000, A. S. Zoble and the author showed that (certain consequences of) Todorcevic's Strong Reflection Principle (SRP) imply ADL(ℝ). (See [22].) Since Martin's Maximum implies SRP, this gave the first derivation of ADL(ℝ) from an “unaugmented” forcing axiom.

In this paper we shall repair some errors and fill some gaps in the inner model theory of [2]. The problems we shall address affect some quite basic definitions and proofs. We shall be concerned with condensation properties of canonical inner models constructed from coherent sequences of extenders as in [2]. Condensation results have the general form: if x is definable in a certain way over a level , then either x ∈ , or else from x we can reconstruct in a simple way. The first condensation property considered in [2] is the initial segment condition, or ISC. In section 1 we show that the version of this condition described in [2] is too strong, in that no coherent in which the extenders are indexed in the manner of [2], and which is such that L[ ] satisfies the mild large cardinal hypothesis that there is a cardinal which is strong past a measurable, can satisfy the full ISC of [2]. It follows that the coherent sequences constructed in [2] do not satisfy the ISC of [2]. We shall describe the weaker ISC which these sequences do satisfy, and indicate the small changes in the arguments of [2] this new condition requires.

In this paper, we shall prove two theorems involving the construction of core models with infinitely many Woodin cardinals. We assume familiarity with [12], which develops core model theory the one Woodin level, and with [10] and [6], which extend the fine structure theory of [5] to mice having many Woodin cardinals. The most important new problem of a general nature which we must face here concerns the iterability of Kc with respect to uncountable iteration trees. Our first result is the following theorem, a slightly stronger version of which was proved independently and earlier by Woodin. The theorem settles positively a conjecture of Feng, Magidor, and Woodin [2]. Theorem. Let Ω be measurable. Then the following are equivalent: (a) for all posets , (b) for every poset , (c) for every poset ℙ ∈ VΩ, Vℙ ⊨ there is no uncountable sequence of distinct reals in L(ℝ) (d) there is an Ω-iterable premouse of height Ω which satisfies “there are infinitely many Woodin cardinals”. It is an immediate corollary that if every set of reals in L(ℝ) is weakly homogeneous, then ADL(ℝ) holds. We shall also indicate some extensions of the theorem to pointclasses beyond L(ℝ), and mice with more than ω Woodin cardinals.

Abstract We show that every sufficiently iterable countable mouse has a unique iteration strategy whose associated iteration maps are lexicographically minimal. This enables us to extend the results of [3] on the good behavior of the standard parameter from tame mice to arbitrary mice.

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2] and below). As a consequence, more powerful large cardinal properties reflect to fine structural inner models. For example, we get the following extension to [MiSt, Theorem 11.3] and [St2, Theorem 0.3]. Suppose that there is a strong cardinal that is a limit of Woodin cardinals. Then there is a good extender sequence such that (1) every level of is a sound, tame mouse, and (2) ⊨ “There is a strong cardinal that is a limit of Woodin cardinals”. Recall that satisfies GCH if all its levels are sound. Another consequence of our work is the following covering property, an extension to [St1, Theorem 1.4] and [St3, Theorem 1.10]. Suppose that fi is a normal measure on Ω and that all premice are tame. Then Kc, the background certified core model, exists and is a premouse of height Ω. Moreover, for μ-almost every α < Ω. Ideas similar to those introduced here allow us to extend the fine structure theory of [Sch] to the level of tame mice. The details of this extension shall appear elsewhere. From the extension of [Sch] and Theorem 0.2, new relative consistency results follow. For example, we have the following application. If there is a cardinal κ such that κ is κ+-strongly compact, then there is a premouse that is not tame.

Let E ⊲ F iff E and F are extenders and E ∈ Ult(V, F). Intuitively, E ⊲ F implies that E is weaker—embodies less reflection—than F. The relation ⊲ was first considered by W. Mitchell in [M74], where it arises naturally in connection with inner models and coherent sequences. Mitchell showed in [M74] that the restriction of ⊲ to normal ultrafilters is well-founded. The relation ⊲ is now known as the Mitchell order, although it is not actually an order. It is irreflexive, and its restriction to normal ultrafilters is transitive, but under mild large cardinal hypotheses, it is not transitive on all extenders. Here is a counterexample. Let κ be (λ + 2)-strong, where λ > κ and λ is measurable. Let E be an extender with critical point κ and let U be a normal ultrafilter with critical point λ such that U ∈ Ult(V, E). Let i: V → Ult(V, U) be the canonical embedding. Then i(E) ⊲ U and U ⊲ E, but by 3.11 of [MS2], it is not the case that i(E) ⊲ E. (The referee pointed out the following elementary proof of this fact. Notice that i ↾ Vλ+2 ∈ Ult(V, E) and X ∈ Ea ⇔ X ∈ i(E)i(a). Moreover, we may assume without loss of generality that = support(E). Thus, if i(E) ∈ Ult(V, E), then E ∈ Ult(V, E), a contradiction.) By going to much stronger extenders, one can show the Mitchell order is not well-founded. The following example is well known. Let j: V → M be elementary, with Vλ ⊆ M for λ = joω(crit(j)). (By Kunen, Vλ+1 ∉ M.) Let E0 be the (crit(j), λ) extender derived from j, and let En+1 = i(En), where i: V → Ult(V, En) is the canonical embedding. One can show inductively that En is an extender over V, and thereby, that En+1 ⊲ En for all n < ω. (There is a little work in showing that Ult(V, En+1) is well-founded.)

Abstract Posner [6] has shown, by a nonuniform proof, that every degree has a complement below 0′. We show that a 1-generic complement for each set of degree between 0 and 0′ can be found uniformly. Moreover, the methods just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above ∅′. In the second half of the paper, we show that the complementation of the degrees below 0′ does not extend to all recursively enumerable degrees. Namely, there is a pair of recursively enumerable degrees a above b such that no degree strictly below a joins b above a. (This result is independently due to S. B. Cooper.) We end with some open problems.

The structure (D, ≤) of the Turing degrees under Turing reducibility is quite complicated. This is true even if we restrict our attention to the substructure (R, ≤) of r.e. degrees. However, the theorems which imply that these structures are complicated all involve ad hoc constructions of sets having the desired reducibility relations. The complexity disappears when we turn to degrees occurring in nature. Of the degrees in R, only 0 and 0′ seem natural. In D, only 0, 0′, 0″, …, 0ω, 0ω+1, … (and on into the transfinite) seem natural. Thus the natural degrees appear to be wellordered, with successors given by Turing jump. If this is true, one would like to prove it. Of course the first problem is to make the concept of naturalness more precise. The following requirements seem plausible: a natural degree should be definable, its definition should relativise to an arbitrary degree, and this relativisation should preserve reducibility relations among natural degrees. Thus to each natural degree c is associated a definable fc: D → D so that fc(0) = c and ∀d(d ≤ fc(d)). Moreover, b ≤ c iff ∀d(fb(d) ≤, fc(d)). To be specific, let us take the definability of fc to mean that fc ∈ L(R). If P is a property of degrees, we say P holds almost everywhere (a.e.) iff ∃c∀d ≥: c P(d). For f, g: D → D, let f ≤mg iff f(d) ≤ g(d) a.e. Define f′ by f′(d) = f{d)′, and let M = {f: D → D/f ∈ L(R) ∧ d ≤ f(d) a.e.}. The following conjecture is due to D. A. Martin: Conjecture. M is prewellordered by ≤m. If f ∈ M has rank α in ≤m, then f′ has rank α + 1.

A pointclass is a class of subsets of the Baire space (ωω) closed under inverse images by continuous functions. The dual of a pointclass Γ, denoted is {~A∣ A ∈ Γ}. (Complements are relative to ωω.) If Γ is nonselfdual, i.e. , then let . We say a nonselfdual pointclass Γ has the first separation property, and write Sep(Γ), iff (∀A, B ∈ Γ)(A ⋂ B = ∅ ⇒ (∃C ∈ Δ)(A ⊆ C ∧ B ⋂ = ∅)). The set C is said to separate A and B. Descriptive set theory abounds in nonselfdual pointclasses Γ, and for the more natural examples of such Γ one can always show by assuming enough determinateness that exactly one of Sep(Γ) and Sep( ) holds. Van Wesep [2] provides a partial explanation of this fact by showing that, assuming the full axiom of determinateness, one of Sep(Γ) and Sep( ) must fail for all nonselfdual pointclasses Γ. We shall complete the explanation by showing that one of Sep(Γ) and Sep( ) must hold. The axiom of determinateness has other interesting consequences in the general theory of pointclasses. See e.g. [3].

Our unexplained notation is that of Rogers [4]. Let P ⊆ 2N × 2N. We call a sequence <An: n ∈ N> of subsets of N a P-sequence iff ∀n(An+1 = the unique B such that P(An, B)). Theorem. Let P ⊆ 2N × 2N be arithmetical. Then there is no P-sequence n: n ∈ N> such that ∀n(A′n+1 ≤T An). This theorem improves a result of Friedman [2] who showed that for no arithmetical P is there a P-sequence <An: n ∈ N> such that An + 1 is a code for an ω-model of the relative arithmetic comprehension schema, and An + 1 is present in the model coded by An, for all n. Other related results are those of Harrison [3], who showed there is a sequence <An: n ∈ N> such that ∀nn + 1 ≤T An>, and of Enderton and Putnam [1], who showed there is no sequence <An: n ∈ N> with ∀n(A′n + 1 ≤T An) and A0 hyperarithmetic. Our theorem is closely connected to Gödel's second incompleteness theorem. Its proof is a recursion theoretic parallel to the proof of Gödel's theorem. In §2 we draw a version of Gödel's theorem as a corollary to ours.